HYP

C. Krattenthaler

Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria. e-mail: KRATT@PAP.UNIVIE.AC.AT

This is a MATHEMATICA package for handling hypergeometric series. It provides quite a few tools for

(A) manipulating factorial expressions

(B) transforming binomial sums into hypergeometric notation (C) summing hypergeometric series (D) transforming hypergeometric series (E) applying contiguous relations (F) doing formal limits of hypergeometric expressions (G) transforming hypergeometric MATHEMATICA expressions into TEX-code. (H) applying Gosper’s [3] and Zeilberger’s [10, 11, 12] algorithms

The tools for items (A), (B), (F), (G) are contained in the file hyp.m, the basic package. This file must be loaded at the very beginning of your MATHEMATICA session. (Ignore error messages occuring when loading hyp.m.) The file hyp.m defines the basic objects, the rules and functions for items (A), (B), (F), and (G), and predefines all the remaining ones. The tools for (C) are the contents of the file summatio.m, those for (D) are the contents of the files transfor.m and transfor.mli, and those for (E) are the contents of the file contig.m. You also have access to summation and transformation formulas in form of equations. This is the contents of the files summatio.mgl and transfor.mgl, respectively. The file output defines some nice MATHEMATICA output features for Sum, Product, Integrate, Abs, Floor, Ceiling, Pi, and Infinity. The tools for item (H) rely on Peter Paule and Markus Schorn’s Mathematica implementation of Gosper’s and Zeilberger’s algorithms. The current version 1.1 or updates can be

received via e-mail request to peter.paule@risc.uni-linz.ac.at.

However, the philosophy of this package is:

Do it by yourself!

The idea is that you should be able to control each step in a series of manipulations by yourself. So, for instance, this package does not make any attempt to sum or transform a series automatically. So, it is you who has to tell the package which command, summation, or transformation has to be applied next. Therefore a basic knowledge of hypergeometric series is required (cf. [1, sec. 1.1, 2.1; 7, sec. 1.1; 2, pp. 1–6]). This handbook provides you with a list of the rules, functions, summations, transformations that are available. The main sources for identities that are included in this package have been the books [2, 7, 1]. They contain a fairly comprehensive collection of known summation and transformation formulas for hypergeometric series. In particular, the (almost) complete Appendices of [7] and [2] (for q ↑ 1) are included in this package.

Finally you should be warned that there is no guarantee that a formula that has been obtained using this package is actually valid. Many formulas or operations are only valid under certain restrictions for the parameters. This package only helps you to do calculations fast. It is up to you to check that the manipulations you are doing are actually being allowed.

For a brief summary of the main features of this package the user is referred to [4] which is the contents of the AMS-TEX file hyp_{–}hypq.tex.

2 The MATHEMATICA package HYP

Hypergeometric notation

All the notation and terminology is adopted from [2, pp. 1–6]. The (generalized) hypergeometric series is defined by

rF_{s}[ \begin{matrix}a_{1}, …, a_{r} \\ b_{1}, …, b_{s} \end{matrix}; z] = ∑_{n=0}^{∞} \frac{(a_{1})_{n} ⋯ (a_{r})_{n}}{n! (b_{1})_{n} ⋯ (b_{s})_{n}} z^{n} ,

where the rising factorial (a)_{n} is given by (a)_{n} := a(a + 1) ⋯ (a + n − 1), n ≥ 1, (a)_{0} := 1. A hypergeometric series r+1F_{r} is called very well-poised if a_{i} + b_{i} = 1 + a_{0} for i = 1, 2, …, r, and among the parameters a_{i} occurs 1 + a_{0}/2. We use the standard abbreviation for very well-poised hypergeometric series,

r+1V_{r}(a_{0}; a_{2}, a_{3}, …, a_{r}; z) := _{r+1}F_{r}[ \begin{matrix}a_{0}, 1 + a_{0}/2, a_{2}, a_{3}, …, a_{r} \\ a_{0}/2, 1 + a_{0} − a_{2}, 1 + a_{0} − a_{3}, …, 1 + a_{0} − a_{r} \end{matrix}; z] .

We also use the compact Gasper-Rahman notation

(a_{1}, a_{2}, …, a_{r})_{n} := (a_{1})_{n} (a_{2})_{n} ⋯ (a_{r})_{n}

and

Γ [ \begin{matrix}a_{1}, a_{2}, …, a_{r} \\ b_{1}, b_{2}, …, b_{s} \end{matrix} ] := \frac{Γ(a_{1}) Γ(a_{2}) ⋯ Γ(a_{r})}{Γ(b_{1}) Γ(b_{2}) ⋯ Γ(b_{s})} .

The file hyp.m

The objects which are defined in the file hyp.m are

AbsGreater, AbsSmaller, AbsUndetermined, Add, AmSLaTeX, AmSTeX, baszerl, baszus, Binomialp, Div,
Drucke, Ers, erw1, erw2, Expandq, F, Factorialp, FCancel, FEinf, FFormat, FOrdne, FPerm, FSum,
FTausche, GAMMA, Gleichung, GlTausche, GOSPER, Gzerl, Gzus, Hoch, hypAttributes, inv, LaTeX, Limes,
lina1, lina2, linz, LS, Mal, ManipulationsListe, MinusOne, neg1, neg2, p, P, paufl, PosListe, PSort,
pzerl, pzus, RS, SchreibeZahl, SimplifyP, Sub, Subst, SumErw1, SumErw2, SumExpand, SumF, SumInfinity,
SumRegeln, SumSammle, SumShift, SumTausche, SumUmkehr, SumZerl, TeX, TeXFV, TeXMat, trans, V, ZB,
zerl, zus1, zus2, zus3.

These objects can be divided into 10 groups: There are the basic objects,

Binomialp, F, Factorialp, GAMMA, p, V,

the rules for manipulating factorial expressions

baszerl, baszus, erw1, erw2, Expandq, Gzerl, Gzus, inv, lina1, lina2, linz, MinusOne, neg1, neg2, paufl,
pzerl, pzus, trans, zerl, zus1, zus2, zus3,

the rules for manipulating sums and hypergeometric series,

FCancel, FEinf, FFormat, FOrdne, FSum, FPerm, FTausche, SumErw1, SumErw2, SumExpand, SumF,
SumInfinity, SumRegeln, SumSammle, SumShift, SumTausche, SumUmkehr, SumZerl,

two functions for controlled use of rules,

Ers, Posliste,

one function for substitution of an expression instead of another expression,

Subst,

some objects for doing limits of hypergeometric expressions,

AbsGreater, AbsSmaller, AbsUndetermined, Limes,

one object for simplifying arguments in hypergeometric expressions,

    SimplifyP,

some objects for converting expressions into TEX-code,

    AmSLaTeX, AmSTeX, LaTeX, TeX, TeXFV, TeXMat,

two objects for on-line help,

    hypAttributes, ManipulationsListe,

and the function

    Drucke,

which enables you to directly send an expression to the printer. Also there are

    Add, Div, Gleichung, GlTausche, Hoch, LS, Mal, PSort, RS, Sub,

for manipulating equations and writing expressions in a “normalized” form (PSort) in order to be able to quickly check if two expressions agree. These objects are particularly important when using objects from summatio.mgl and transfor.mgl. Finally there are

    GOSPER, ZB,

for applying Gosper’s and Zeilberger’s algorithms. Peter Paule’s and Markus Schorn’s Mathematica implementation of these algorithms which is needed for using GOSPER and ZB also provides

    Zb, Gosper, RunMode, FileName, SolAmount, Fnk, GoRat, GoSol, Cert, DegBound, System, SystemDimension.

Regarding these objects the user is referred to the documentation and the description [5] of this implementation.

Most of the tools for manipulating expressions that are provided by this package are rules. This has the advantage that very often you do not have to specify to which part of an expression you want to apply a rule, since there is just one subexpression to which the rule applies. However, if there are more subexpressions to which a rule applies, you will sometimes want to apply the rule only to some of the subexpressions. To handle this conveniently, there are the functions Ers and PosListe.

                                The file summatio.m

This file basically contains the Appendix II of [2] for q ↑ 1 in form of rules. You do not have to load this file by hand since it is loaded automatically once an object of this file is called. The objects that are defined by summatio.m are

    S1001, S2101, S2103, S2104, S2105, S2106, S2131, S2132, S3201, S3202, S3204, S3231, S3232, S3233,
    S3234, S3235, S3261, S3291, S4306, S4307, S4331, S4332, S5431, S5432, S6531, S6532, S7631, S7632,
    S7691, SListe, SumListe.

The numbering of each rule is S⟨d1⟩⟨d2⟩⟨n1⟩ following the following system: The number ⟨d1⟩ is the number of the upper parameters, the number ⟨d2⟩ is the number of the lower parameters of the hypergeometric series to which the rule applies. The number ⟨n1⟩ allows to distinguish the rules applying to hypergeometric series with equal numbers of upper and lower parameters. ⟨n1⟩ is within the range 01–30 if the summation is a one-term summation and its qanalogue (in the package HYPQ) has the same numbering, it is within the range 31–60 if the summation is a one-term summation and its q-analogue (in the package HYPQ) has a different numbering, it is within the range 61–90 if the summation is a two- or more-term summation and its q-analogue (in the package HYPQ) has the same numbering, it is within the range 90–120 if the summation is a two- or more-term summation and its q-analogue (in the package HYPQ) has a different numbering.

For terminating series there is a check if one of the parameters is of the form (−n) where n is a nonnegative integer. Depending on your input you might be asked if some expression is a nonnegative integer (see the examples for S3201). Be sure to give an affirmative answer only for one of several expressions, otherwise the package will try to find the minimum of all of these, which might cause problems. This remark also applies to other rules which put this question, e.g. SumUmkehr, FSum, or in case that automatic evaluating is active (cf. P). The rule SListe enables you to quickly check if one of the summation rules S1001–S7691 can be directly applied.

                            The file transfor.m

This file basically contains the Appendix III of [2] for q ↑ 1 in form of rules. You do not have to load this file by hand since it is loaded automatically once an object of this file is called. The objects that are defined by transfor.m are

T2103, T2104, T2106, T2107, T2110, T2112, T2131, T2132, T2133, T2134, T2135, T2136, T2137, T2138,
T2139, T2139, T2140, T2141, T2163, T2191, T2192, T3204, T3205, T3206, T3207, T3217, T3231, T3232,
T3233, T3234, T3235, T3236, T3237, T3238, T3239, T3240, T3261, T3262, T3263, T3264, T4301, T4302,
T4303, T4304, T4306, T4309, T4310, T4312, T4331, T4332, T4362, T4391, T5401, T5402, T5403, T5468,
T6501, T6531, T6532, T6533, T6534, T7631, T7632, T7633, T7634, T7635, T7636, T7637, T7691, T7692,
T7693, T7694, T8731, T8732, T9831, T9832, T9833, T9834, T9835, T9836, T9837, T9891, T9892, T9893,
T9894, T111031, T111032, TListe, TransListe.

The comments for the file summatio.m regarding the numbering of the rules and optional questions for input also apply here. The rule TListe enables you to quickly check if one of the transformation rules T2103–T9891 can be directly applied.

                           The file transfor.mli

Each of the objects of this file corresponds to a transformation rule of the file transfor.m. Each object gives a list of all the outcomes under application of a particular transformation after before having permuted the upper and lower parameters of the involved basic hypergeometric series. All the objects in this file are rules. These rules help to prove conjectured transformation formulas quickly. You do not have to load this file by hand since it is loaded automatically once an object of this file is called. The objects that are defined by transfor.mli are

Tli2103, Tli2104, Tli2106, Tli2107, Tli2110, Tli2112, Tli2131, Tli2132, Tli2133, Tli2134, Tli2135,
Tli2136, Tli2137, Tli2138, Tli2139, Tli2139, Tli2140, Tli2141, Tli2163, Tli2191, Tli2192, Tli3204,
Tli3205, Tli3206, Tli3207, Tli3217, Tli3231, Tli3232, Tli3233, Tli3234, Tli3235, Tli3236, Tli3237,
Tli3238, Tli3239, Tli3240, Tli3261, Tli3262, Tli3263, Tli3264, Tli4301, Tli4302, Tli4303, Tli4304,
Tli4306, Tli4309, Tli4310, Tli4312, Tli4331, Tli4332, Tli4362, Tli4391, Tli5401, Tli5402, Tli5403,
Tli5468, Tli6501, Tli6531, Tli6532, Tli6533, Tli6534, Tli7631, Tli7632, Tli7633, Tli7634, Tli7635,
Tli7636, Tli7637, Tli7691, Tli7692, Tli7693, Tli7694, Tli8731, Tli8732, Tli9831, Tli9832, Tli9833,
Tli9834, Tli9835, Tli9836, Tli9837, Tli9891, Tli9892, Tli9893, Tli9894, Tli111031, Tli111032.

                   The files summatio.mgl and transfor.mgl

These files contain the same summations, respectively transformations, as summatio.m, respectively transfor.m, but in form of equations. You do not have to load these files by hand since they are loaded automatically once an object of this file is called. The respective objects are

Sgl1001, Sgl2101, Sgl2103, Sgl2104, Sgl2105, Sgl2106, Sgl2131, Sgl2132, Sgl3201, Sgl3202, Sgl3204,
Sgl3231, Sgl3232, Sgl3233, Sgl3234, Sgl3235, Sgl3261, Sgl3291, Sgl4306, Sgl4307, Sgl4331, Sgl4332,
Sgl5431, Sgl5432, Sgl6531, Sgl6532, Sgl7631, Sgl7632, Sgl7691, SumListe$gl,

and

Tgl2103, Tgl2104, Tgl2106, Tgl2107, Tgl2110, Tgl2112, Tgl2131, Tgl2132, Tgl2133, Tgl2134, Tgl2135,
Tgl2136, Tgl2137, Tgl2138, Tgl2139, Tgl2139, Tgl2140, Tgl2141, Tgl2163, Tgl2191, Tgl2192, Tgl3204,
Tgl3205, Tgl3206, Tgl3207, Tgl3217, Tgl3231, Tgl3232, Tgl3233, Tgl3234, Tgl3235, Tgl3236, Tgl3237,
Tgl3238, Tgl3239, Tgl3240, Tgl3261, Tgl3262, Tgl3263, Tgl3264, Tgl4301, Tgl4302, Tgl4303, Tgl4304,
Tgl4306, Tgl4309, Tgl4310, Tgl4312, Tgl4331, Tgl4332, Tgl4362, Tgl4391, Tgl5401, Tgl5402, Tgl5403,
Tgl5468, Tgl6501, Tgl6531, Tgl6532, Tgl6533, Tgl6534, Tgl7631, Tgl7632, Tgl7633, Tgl7634, Tgl7635,
Tgl7636, Tgl7637, Tgl7691, Tgl7692, Tgl7693, Tgl7694, Tgl8731, Tgl8732, Tgl9831, Tgl9832, Tgl9833,
Tgl9834, Tgl9835, Tgl9836, Tgl9837, Tgl9891, Tgl9892, Tgl9893, Tgl9894, Tgl111031, Tgl111032,
TransListe$gl.

When calling one of these objects you will be put a question. If the variables of the called summation or transformation are undefined, the question is

Do you want to set values for the equation?  [y|n]:

Enter y if you want to set values, even only for some of them, if you do not need to set values enter n. If some of the variables of the called summation or transformation are already defined, you will be asked the question

Some variables have a value.  Should the variables
{[V,a,r,i,a,b,l,e,s]} be cleared?  Do you want to set
values for the equation (v)?  [y|n|yv|nv]:

Now you have four options depending on if you want to set values or not and if you want to clear the already defined variables or not. For example, if you want to set values but do not want to clear the defined variables, enter nv. (Cf. the examples in Sgl2101.)

In addition there are the functions and variables

    Add, Div, Gleichung, GlTausche, Hoch, LS, Mal, RS, Sub,

for manipulating equations. In fact, once you have called one of the objects Sgl∗ or Tgl∗, the right-hand side of the displayed equation will have been assigned to RS, the left-hand side to LS, and thus the equation itself to Gleichung (cf. the example in Gleichung). The functions Add, Div, GlTausche, Hoch, Mal, Sub, and also Ers, allow you to manipulate the equation.

                                 The file contig.m

This file contains a vast number of contiguous relations in form of rules. You do not have to load this file by hand since it is loaded automatically once an object of this file is called. The objects that are defined by contig.m are

    C01, C02, C14, C15, C16, C17, C18, C19, C20, C21, C22, C23, C24, C25, C26, C27, C30, C31, C32, C33, C34, C35,
    C36, C40, C41, C42, C43, C44, C45, C46, C49, C50, C51, C52, C53, C54, C55, C56, C57, C58, C59, C60, C61, C62,
    C63, C64, C65, C66, C67, C68, C69, C70, C71, C72, C73, C74, C75, C76, C77, C78, C79, C80, C81, C82, C83, C84,
    C85, C86, C87, C88, C89, C90, C91, C92, C93, C94, C95, C96, C97, C98, C99, C100, C101, C102, C103, C104,
    C105, C106, C107, C108, C109, C110, C111, C112, C113, C114, C115, C116, C117, C118, C119, C120, C121,
    ContigListe.

Simultaneous use of HYP and HYPQ

It is possible to load both packages, HYP and HYPQ. In this case, the objects of the package that is loaded last will override those objects of the other package which have identical names. However, you can use the overrided objects by calling them by their full names. To determine the full name of an object the following rule applies:

If the object Object is defined in the file File.ext, then the full name of Object is File`ext`Object.

For instance, if you load hyp.m first and then hyp.q and want to use Limes with an ordinary (i.e. non-basic) hypergeometric expression, then you have to type Hyp`m`Limes instead of Limes. (Calling Limes would invoke the basic hypergeometric Limes.) For information on contexts in MATHEMATICA confer [9].

On-line help

For each object of this package on-line help is supported in the usual way. For instance, quick information about Limes (not having the manual at hand) is available in the following way.

In[1]:= ?Limes
Description:  Function for doing formal limits of hypergeometric
  expressions.  If required for taking the limit, you will be asked whether
  or not the absolute value of some variable or expression is smaller than
  1.  Your decision is stored for the rest of your MATHEMATICA session.  If
  you want to change your decision later, use "AbsGreater", "AbsSmaller",
  or "AbsUndetermined", respectively.
Warning:  This function uses primitive algebraic techniques to do the
  limit.  There is no check if taking the limit is actually allowed.  So it
  is left to you to check the validity of a result of "Limes".
Usage:  Limes[Expr, x-⟩x0].
See also:  AbsGreater, AbsSmaller, AbsUndetermined, MinusOne.

The screen output

The screen output of the examples in this manual imitates the output under usage of code tables 437, 860, 863, or 865 (cf. the read.me). The output under other code tables is a little bit less attractive, but similar. For instance, the examples for GAMMA then would read as follows.

In[1]:= GAMMA[2*a+1]
Out[1]= Ga(1 + 2 a)
In[2]:= GAMMA[{a,b},{c,d}]
          [      ]
          | a, b |
Out[2]= Ga|      |
          | c, d |
          [      ]

The MATHEMATICA package HYP 7

A brief dictionary

Most of the names of the objects are obviously German influenced. To help those who are not so familiar with German, brief German–English and English–German vocabularies are provided.

A German–English vocabulary

German  English  Mathematica objects in HYP

containing the word

abspalten split  lina1, lina2
auflösen dissolve  paufl
drucken print Drucke
einfügen insert  FEinf
ersetzen replace  Ers

erweitern extend  erw1, erw2, SumErw1, SumErw2

Gleichung equation  Gleichung
”2 hoch 3” ”2 to the 3”  Hoch
”2 mal 3” ”2 times 3”  Mal
ordnen  order  FOrdne

Regel  rule  SumRegeln

sammeln collect  SumSammle
schreiben write  SchreibeZahl
tauschen exchange, interchange SumTausche, GlTausche
umkehren reverse  SumUmkehr

Zahl  number  SchreibeZahl
zerlegen split  zerl1, pzerl, Gzerl, SumZerl

zusammen together  zus1, zus2, zus3

8 The MATHEMATICA package HYP

An English–German vocabulary

English  German  Mathematica objects in HYP

containing the word

collect  sammeln SumSammle
dissolve  auflösen paufl
equation  Gleichung Gleichung

exchange  tauschen SumTausche, GlTausche
extend  erweitern erw1, erw2, SumErw1, SumErw2

insert  einfügen FEinf

interchange tauschen SumTausche, GlTausche
number  Zahl  SchreibeZahl

order  ordnen  FOrdne
print  drucken  Drucke
replace  ersetzen Ers
reverse  umkehren SumUmkehr

rule  Regel  SumRegeln

split  abspalten lina1, lina2

split  zerlegen zerl1, pzerl, Gzerl, SumZerl

”2 times 3” ”2 mal 3” Mal
”2 to the 3” ”2 hoch 3” Hoch
together  zusammen zus1, zus2, zus3
write  schreiben SchreibeZahl

Alphabetic List of the objects with desriptions

AbsGreater \underline{Description}: Function for declaring the absolute value of a variable or expression to be greater than 1. This declaration

is used by Limes.

\underline{Usage}: AbsGreater[Expr].
\underline{See also}: AbsSmaller, AbsUndetermined, Limes.

AbsSmaller \underline{Description}: Function for declaring the absolute value of a variable or expression to be smaller than 1. This declaration

is used by Limes.

\underline{Usage}: AbsSmaller[Expr].
\underline{See also}: AbsGreater, AbsUndetermined, Limes.

AbsUndetermined \underline{Description}: Function for declaring the absolute value of a variable or expression to be neither smaller nor greater

than 1. This declaration is used by Limes.

\underline{Usage}: AbsUndetermined[Expr].
\underline{See also}: AbsGreater, AbsSmaller, Limes.

Add \underline{Description}: Function that adds Expr to Gleichung.

\underline{Usage}: Add[Expr].

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
                           (-a + c)
             a, -n                 n
Out[1]=  F        ; 1   == ---------
        2 1    c             (c)
                                n
In[2]:= Add[p[c-a,n]]
                                                   (-a + c)
             a, -n                                         n
Out[2]=  F        ; 1   + (-a + c)  == (-a + c)  + ---------
        2 1    c                  n            n     (c)
                                                        n

In[3]:= Gleichung
                                                   (-a + c)
             a, -n                                         n
Out[3]=  F        ; 1   + (-a + c)  == (-a + c)  + ---------
        2 1    c                  n            n     (c)
                                                        n
\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Mal, Div, Sub, Hoch, GlTausche, Ers.

AmSLaTeX

\underline{Description}: Switch that changes the output of TeXForm to be usable with AMS-LaTEX. By default the output of TeXForm is usable with AMS-TEX.

\underline{Usage}: AmSLaTeX.

\underline{Example(s)}:

In[1]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.
In[2]:= TeXForm[F[{a,b},{c},z]]
Out[2]//TeXForm=
  {} {2} F {1} \!\left [ \matrix { a, b}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]
In[3]:= AmSLaTeX
In[4]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-LaTeX.
TeXForm uses V[] for very well-poised hypergeometric series.
In[5]:= TeXForm[F[{a,b},{c},z]]
Out[5]//TeXForm=
  {} {2} F {1} \!\left [ \begin{matrix} { a, b}\\ { c}\end{matrix} ;
   {\displaystyle z}\right ]
\underline{See also}: AmSTeX, LaTeX, TeX, TeXMat, TeXFV.

Alphabetic List: baszerl 11

AmSTeX \underline{Description}: Switch that changes the output of TeXForm to be usable with AMS-TEX. By default the output of

TeXForm is usable with AMS-TEX.

\underline{Usage}: AmSTeX.

\underline{Example(s)}:

In[1]:= TeX
In[2]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses V[] for very well-poised hypergeometric series.
In[3]:= TeXForm[F[{a,b},{c},z]]
Out[3]//TeXForm=
  {} {2} F {1} \!\left [ \matrix { a, b\cr c} ; {\displaystyle z} \right ]
In[4]:= AmSTeX
In[5]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.
In[6]:= TeXForm[F[{a,b},{c},z]]
Out[6]//TeXForm=
  {} {2} F {1} \!\left [ \matrix { a, b}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]
\underline{See also}: AmSLaTeX, LaTeX, TeX, TeXMat, TeXFV.

baszerl

\underline{Description}: \begin{matrix}(a)_{n} → m^{n} ∏_{k=0}^{m−1}((a + k)/m)_{n/m}, \\ Γ(a) → m^{a−1/2}(2π)^{(1−m)/2} ∏_{k=0}^{m−1} Γ((a + k)/m). \end{matrix}

The parameter m has to be entered on request.

\underline{Usage}: Expr/.baszerl.

\underline{Example(s)}:

In[1]:= p[a,n]

Out[1]= (a)
           n
In[2]:= %/.baszerl
split into ?  terms:  2
         n  a      1 + a
Out[2]= 2  (-)    (-----)
            2 n/2    2   n/2
In[3]:= p[b,4*m]
Out[3]= (b)
           4 m
In[4]:= %/.baszerl
split into ?  terms:  4
         4 m  b    1 + b    2 + b    3 + b
Out[4]= 4    (-)  (-----)  (-----)  (-----)
              4 m    4   m    4   m    4   m
In[5]:= GAMMA[2*c]
Out[5]= Γ(2 c)
In[6]:= %/.baszerl
split into ?  terms:  4
         -(1/2) + 2 c   c    1 + 2 c    2 + 2 c    3 + 2 c
        4  Γ(-) Γ(-------) Γ(-------) Γ(-------)
                        2       4          4          4
Out[6]= ---------------------------------------------------
                              3/2  3/2
                             2  π
\underline{See also}: baszerl, baszus, Ers, PosListe, ManipulationsListe.

baszus

\underline{Description}: \begin{matrix}(a)_{n} → m^{−mn}(am)_{mn}/ ∏_{k=1}^{m−1}(a + k/m)_{n}, \\ Γ(a) → m^{1/2−am}(2π)^{(m−1)/2}Γ(am)/ ∏_{k=1}^{m−1} Γ(a + k/m). \end{matrix}

The parameter m has to be entered on request. This operation is basically the inverse of baszerl.

\underline{Usage}: Expr/.baszus.

\underline{Example(s)}:

In[1]:= p[a/2,n]*p[(a+1)/2,n]

         a    1 + a
Out[1]= (-)  (-----)
         2 n    2   n
In[2]:= Ers[%,baszus,{1}]
put together ?  terms:  2
                1 + a
        (a)    (-----)
           2 n    2   n
Out[2]= ---------------
          2 n  1   a
         2    (- + -)
               2   2 n
In[3]:= ExpandAll[%]
        (a)
           2 n
Out[3]= ------
          2 n
         2
In[4]:= GAMMA[(a-1)/2]*GAMMA[a/2]*p[b/3,m]*p[(b+1)/3,m]*p[(b+2)/3,m]
          -1 + a    a   b    1 + b    2 + b
Out[4]= Γ(------) Γ(-) (-)  (-----)  (-----)
            2       2   3 m    3   m    3   m
In[5]:= Ers[%,baszus,{3}]
put together ?  terms:  3
          -1 + a    a          1 + b    2 + b
        Γ(------) Γ(-) (b)    (-----)  (-----)
            2       2     3 m    3   m    3   m
Out[5]= ---------------------------------------
                 3 m  1   b    2   b
                3    (- + -)  (- + -)
                      3   3 m  3   3 m
In[6]:= ExpandAll[%]
            1    a    a
        Γ(-(-) + -) Γ(-) (b)
            2    2    2     3 m
Out[6]= -----------------------
                  3 m

In[7]:= PosListe[%]
           -3 m               1    a              a
Out[7]= {{3    , {{1}}}, {Γ(-(-) + -), {{2}}}, {Γ(-), {{3}}}, {(b)   , {{4}}}}
                              2    2              2               3 m
In[8]:= Ers[%%,baszus,{2}]
put together ?  terms:  2
         1 - 2 (-(1/2) + a/2)                1    a
        2                     Sqrt[π] Γ(2 (-(-) + -)) (b)
                                             2    2      3 m
Out[8]= ----------------------------------------------------
                                 3 m
                                3
In[9]:= ExpandAll[%]

        4 Sqrt[π] Γ(-1 + a) (b)

                               3 m
Out[9]= --------------------------
                  a  3 m
                 2  3

\underline{See also}: Ers, PosListe, ManipulationsListe.

Binomialp

\underline{Description}: Binomialp[n,k] is the binomial coefficient, written in terms of factorial symbols (Pochhammer symbols)

  p.
\underline{Usage}: Binomialp[n,k].

\underline{Example(s)}:

In[1]:= Binomialp[n,k]
        (1 - k + n)
                   k
Out[1]= ------------
            (1)
               k
In[2]:= Binomialp[6,3]

Alphabetic List: C14 15

        (4)
           3
Out[2]= ----
        (1)
           3
\underline{See also}: Factorialp.

C01 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ (B) \end{matrix} ; z] ⟶ 1 + z \frac{∏_{i=1}^{r} A_{i}}{∏_{i=1}^{s} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}1, (A + 1) \\ 2, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C01.
\underline{See also}: C64, ContigListe, Ers, PosListe.

C02 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A), 1 \\ (B) \end{matrix} ; z] ⟶ \frac{1}{z} \frac{∏_{i=1}^{s}(B_{i} − 1)}{∏_{i=1}^{r}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}(A − 1), 1 \\ (B − 1) \end{matrix} ; z] − \frac{1}{z} \frac{∏_{i=1}^{s}(B_{i} − 1)}{∏_{i=1}^{r}(A_{i} − 1)}

\underline{Usage}: Expr/.C02.
\underline{See also}: C64, ContigListe, Ers, PosListe.

C14 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a − 1, (A) \\ (B) \end{matrix} ; z] + z \frac{∏_{i=1}^{r−1} A_{i}}{∏_{i=1}^{s} B_{i}}_{r}F_{s}[ \begin{matrix}a, (A + 1) \\ (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C14[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C15 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a + 1, (A) \\ (B) \end{matrix} ; z] − z \frac{∏_{i=1}^{r−1} A_{i}}{∏_{i=1}^{s} B_{i}}_{r}F_{s}[ \begin{matrix}a + 1, (A + 1) \\ (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C15[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C16 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; z] ⟶ \frac{1}{z} \frac{∏_{i=1}^{s}(B_{i} − 1)}{∏_{i=1}^{r−1}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}a, (A − 1) \\ (B − 1) \end{matrix} ; z] − \frac{1}{z} \frac{∏_{i=1}^{s}(B_{i} − 1)}{∏_{i=1}^{r−1}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}a − 1, (A − 1) \\ (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C16[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C17 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a − 1, (A) \\ (B) \end{matrix} ; z] + z \frac{∏_{i=1}^{r−1} A_{i}}{∏_{i=1}^{s} B_{i}}_{r}F_{s}[ \begin{matrix}a, (A + 1) \\ (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C17[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C18 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a + 1, (A) \\ (B) \end{matrix} ; z] − z \frac{∏_{i=1}^{r−1} A_{i}}{∏_{i=1}^{s} B_{i}}_{r}F_{s}[ \begin{matrix}a + 1, (A + 1) \\ (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C18[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; z] ⟶ \frac{1}{z} \frac{∏_{i=1}^{s}(B_{i} − 1)}{∏_{i=1}^{r−1}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}a, (A − 1) \\ (B − 1) \end{matrix} ; z] − \frac{1}{z} \frac{∏_{i=1}^{s}(B_{i} − 1)}{∏_{i=1}^{r−1}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}a − 1, (A − 1) \\ (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C19[m1].

m1 is the position of the special upper parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C20 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}(A) \\ b + 1, (B) \end{matrix} ; z] + \frac{z}{b (1 + b)} \frac{∏_{i=1}^{r} A_{i}}{∏_{i=1}^{s−1} B_{i}}_{r}F_{s}[ \begin{matrix}(A + 1) \\ b + 2, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C20[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C21 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}(A) \\ b − 1, (B) \end{matrix} ; z] − \frac{z}{(b − 1) b} \frac{∏_{i=1}^{r} A_{i}}{∏_{i=1}^{s−1} B_{i}}_{r}F_{s}[ \begin{matrix}(A + 1) \\ b + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C21[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C22 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; z] ⟶ \frac{(b − 2) (b − 1)}{z} \frac{∏_{i=1}^{s−1}(B_{i} − 1)}{∏_{i=1}^{r}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}(A − 1) \\ b − 2, (B − 1) \end{matrix} ; z]

− \frac{(b − 2) (b − 1)}{z} \frac{∏_{i=1}^{s−1}(B_{i} − 1)}{∏_{i=1}^{r}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}(A − 1) \\ b − 1, (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C22[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}(A) \\ b + 1, (B) \end{matrix} ; z] + \frac{z}{b (1 + b)} \frac{∏_{i=1}^{r} A_{i}}{∏_{i=1}^{s−1} B_{i}}_{r}F_{s}[ \begin{matrix}(A + 1) \\ b + 2, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C23[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C24 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}(A) \\ b − 1, (B) \end{matrix} ; z] − \frac{z}{(b − 1) b} \frac{∏_{i=1}^{r} A_{i}}{∏_{i=1}^{s−1} B_{i}}_{r}F_{s}[ \begin{matrix}(A + 1) \\ b + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C24[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C25 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ b, (B) \end{matrix} ; z] ⟶ \frac{(b − 2) (b − 1)}{z} \frac{∏_{i=1}^{s−1}(B_{i} − 1)}{∏_{i=1}^{r}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}(A − 1) \\ b − 2, (B − 1) \end{matrix} ; z]

− \frac{(b − 2) (b − 1)}{z} \frac{∏_{i=1}^{s−1}(B_{i} − 1)}{∏_{i=1}^{r}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}(A − 1) \\ b − 1, (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C25[n1].

n1 is the position of the special lower parameter.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C26 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; z] ⟶ \frac{b}{b − a}_{r}F_{s}[ \begin{matrix}a, b + 1, (A) \\ (B) \end{matrix} ; z] + \frac{a}{a − b}_{r}F_{s}[ \begin{matrix}a + 1, b, (A) \\ (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C26[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; z] ⟶ \frac{(a − b − 1)}{a − 1}_{r}F_{s}[ \begin{matrix}a − 1, b, (A) \\ (B) \end{matrix} ; z] + \frac{b}{a − 1}_{r}F_{s}[ \begin{matrix}a − 1, b + 1, (A) \\ (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C27[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C30 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a − 1, b + 1, (A) \\ (B) \end{matrix} ; z] + (1 − a + b) z \frac{∏_{i=1}^{r−2} A_{i}}{∏_{i=1}^{s} B_{i}}_{r}F_{s}[ \begin{matrix}a, b + 1, (A + 1) \\ (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C30[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C31 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; z] ⟶ \frac{1}{(b − a) z} \frac{∏_{i=1}^{s}(B_{i} − 1)}{∏_{i=1}^{r−2}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}a, b − 1, (A − 1) \\ (B − 1) \end{matrix} ; z] − \frac{1}{(b − a) z} \frac{∏_{i=1}^{s}(B_{i} − 1)}{∏_{i=1}^{r−2}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}a − 1, b, (A − 1) \\ (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C31[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C32 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a − 1, b − 1, (A) \\ (B) \end{matrix} ; z] + (a + b − 1) z \frac{∏_{i=1}^{r−2} A_{i}}{∏_{i=1}^{s} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a, b, a + b, (A + 1) \\ −1 + a + b, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C32[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a + 1, b + 1, (A) \\ (B) \end{matrix} ; z] − (1 + a + b) z \frac{∏_{i=1}^{r−2} A_{i}}{∏_{i=1}^{s} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a + 1, b + 1, a + b + 2, (A + 1) \\ a + b + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C33[m1,m2].

m1, m2 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C34 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; z] ⟶ \frac{(b − 1)}{b − 1 − a}_{r}F_{s}[ \begin{matrix}a, (A) \\ b − 1, (B) \end{matrix} ; z] + \frac{a}{1 + a − b}_{r}F_{s}[ \begin{matrix}a + 1, (A) \\ b, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C34[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C35 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; z] ⟶ \frac{(b − 1)}{a − 1}_{r}F_{s}[ \begin{matrix}a − 1, (A) \\ b − 1, (B) \end{matrix} ; z] + \frac{(a − b)}{a − 1}_{r}F_{s}[ \begin{matrix}a − 1, (A) \\ b, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C35[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C36 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; z] ⟶ \frac{(b − a)}{b}_{r}F_{s}[ \begin{matrix}a, (A) \\ 1 + b, (B) \end{matrix} ; z] + \frac{a}{b}_{r}F_{s}[ \begin{matrix}a + 1, (A) \\ b + 1, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C36[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C40 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a − 1, (A) \\ b − 1, (B) \end{matrix} ; z] + \frac{(b − a) z}{(b − 1) b} \frac{∏_{i=1}^{r−1} A_{i}}{∏_{i=1}^{s−1} B_{i}}_{r}F_{s}[ \begin{matrix}a, (A + 1) \\ b + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C40[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a + 1, (A) \\ b + 1, (B) \end{matrix} ; z] − \frac{(b − a) z}{b (1 + b)} \frac{∏_{i=1}^{r−1} A_{i}}{∏_{i=1}^{s−1} B_{i}}_{r}F_{s}[ \begin{matrix}a + 1, (A + 1) \\ b + 2, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C41[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C42 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; z] ⟶ \frac{(b − 2) (b − 1)}{(b − a − 1) z} \frac{∏_{i=1}^{s−1}(B_{i} − 1)}{∏_{i=1}^{r−1}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}a, (A − 1) \\ b − 1, (B − 1) \end{matrix} ; z]

− \frac{(b − 2) (b − 1)}{(b − a − 1) z} \frac{∏_{i=1}^{s−1}(B_{i} − 1)}{∏_{i=1}^{r−1}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}a − 1, (A − 1) \\ b − 2, (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C42[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C43 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a − 1, (A) \\ b + 1, (B) \end{matrix} ; z] + \frac{(a + b) z}{b (1 + b)} \frac{∏_{i=1}^{r−1} A_{i}}{∏_{i=1}^{s−1} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a, 1 + a + b, (A + 1) \\ b + 2, a + b, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C43[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C44 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a + 1, (A) \\ b − 1, (B) \end{matrix} ; z] − \frac{(a + b) z}{(b − 1) b} \frac{∏_{i=1}^{r−1} A_{i}}{∏_{i=1}^{s−1} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}1 + a, a + b + 1, (A + 1) \\ b + 1, a + b, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C44[m1,n1].

m1 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; z] ⟶ \frac{(a − 1)}{a − b}_{r}F_{s}[ \begin{matrix}(A) \\ −1 + a, b, (B) \end{matrix} ; z] + \frac{(b − 1)}{b − a}_{r}F_{s}[ \begin{matrix}(A) \\ a, −1 + b, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C45[n1,n2].

n1, n2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C46 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; z] ⟶ \frac{(b − 1)}{a}_{r}F_{s}[ \begin{matrix}(A) \\ a + 1, b − 1, (B) \end{matrix} ; z] + \frac{(1 + a − b)}{a}_{r}F_{s}[ \begin{matrix}(A) \\ a + 1, b, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C46[n1,n2].

n1, n2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C49 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}(A) \\ a + 1, b − 1, (B) \end{matrix} ; z] + \frac{(b − a − 1) z}{a (1 + a) (b − 1) b} \frac{∏_{i=1}^{r} A_{i}}{∏_{i=1}^{s−2} B_{i}}_{r}F_{s}[ \begin{matrix}(A + 1) \\ a + 2, b + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C49[n1,n2].

n1, n2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C50 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; z] ⟶ \frac{(a − 2) (a − 1) (b − 2) (b − 1)}{(b − a) z} \frac{∏_{i=1}^{s−2}(B_{i} − 1)}{∏_{i=1}^{r}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}(A − 1) \\ −2 + a, b − 1, (B − 1) \end{matrix} ; z]

− \frac{(a − 2) (a − 1) (b − 2) (b − 1)}{(b − a) z} \frac{∏_{i=1}^{s−2}(B_{i} − 1)}{∏_{i=1}^{r}(A_{i} − 1)}_{r}F_{s}[ \begin{matrix}(A − 1) \\ a − 1, b − 2, (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C50[n1,n2].

n1, n2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}(A) \\ a + 1, b + 1, (B) \end{matrix} ; z]+\frac{(1 + a + b) z}{a (1 + a) b (1 + b)} \frac{∏_{i=1}^{r} A_{i}}{∏_{i=1}^{s−2} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a + b + 2, (A + 1) \\ a + 2, b + 2, a + b + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C51[n1,n2].

n1, n2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C52 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}(A) \\ a − 1, b − 1, (B) \end{matrix} ; z]

− \frac{(a + b − 1) z}{(a − 1) a (b − 1) b} \frac{∏_{i=1}^{r} A_{i}}{∏_{i=1}^{s−2} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a + b, (A + 1) \\ a + 1, 1 + b, b − a + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C52[n1,n2].

n1, n2 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C53 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, c, (A) \\ (B) \end{matrix} ; z] ⟶ \frac{b (c − a − 1)}{(b − a) (c − 1)}_{r}F_{s}[ \begin{matrix}a, b + 1, c − 1, (A) \\ (B) \end{matrix} ; z] + \frac{a (c − b − 1)}{(a − b) (c − 1)}_{r}F_{s}[ \begin{matrix}a + 1, b, c − 1, (A) \\ (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C53[m1,m2,m3].

m1, m2, m3 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C54 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, (A) \\ c, (B) \end{matrix} ; z] ⟶ \frac{b (c − a)}{(b − a) c}_{r}F_{s}[ \begin{matrix}a, b + 1, (A) \\ c + 1, (B) \end{matrix} ; z] + \frac{a (c − b)}{(a − b) c}_{r}F_{s}[ \begin{matrix}a + 1, b, (A) \\ c + 1, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C54[m1,m2,n1].

m1, m2 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, b, (A) \\ c, (B) \end{matrix} ; z] ⟶ \begin{matrix}(1 − a + b) & (c − 1) \\ (a − 1) & (1 + b − c) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, b, (A) \\ c − 1, (B) \end{matrix} ; z] + \frac{b (c − a)}{(a − 1) (c − b − 1)}_{r}F_{s}[ \begin{matrix}a − 1, b + 1, (A) \\ c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C55[m1,m2,n1].

m1, m2 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C56 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ b, c, (B) \end{matrix} ; z] ⟶ \frac{(b − 1) (c − a)}{(b − a − 1) c}_{r}F_{s}[ \begin{matrix}a, (A) \\ b − 1, c + 1, (B) \end{matrix} ; z] + \frac{a (1 − b + c)}{(1 + a − b) c}_{r}F_{s}[ \begin{matrix}a + 1, (A) \\ b, c + 1, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C56[m1,n1,n2].

m1 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C57 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ b, c, (B) \end{matrix} ; z] ⟶ \begin{matrix}(b − 1) & (c − a) \\ (a − 1) & (c − b) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, (A) \\ b − 1, c, (B) \end{matrix} ; z] + \begin{matrix}(b − a) & (c − 1) \\ (a − 1) & (b − c) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, (A) \\ b, c − 1, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C57[m1,n1,n2].

m1 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C58 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ a, b, c, (B) \end{matrix} ; z] ⟶ \frac{(a − 1) (1 − b + c)}{(a − b) c}_{r}F_{s}[ \begin{matrix}(A) \\ a − 1, b, c + 1, (B) \end{matrix} ; z]

+ \frac{(b − 1) (1 − a + c)}{(b − a) c}_{r}F_{s}[ \begin{matrix}(A) \\ a, b − 1, c + 1, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C58[n1,n2,n3].

n1, n2, n3 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

Alphabetic List: C62 25

C59 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, 2 − a + c, b, c − b, (A) \\ (B) \end{matrix} ; z] ⟶ \begin{matrix}(a − b − 1) & (1 − a − b + c) \\ (a − 1) & (1 − a + c) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, 1 − a + c, b, c − b, (A) \\ (B) \end{matrix} ; z]

+ \frac{b (c − b)}{(a − 1) (1 − a + c)}_{r}F_{s}[ \begin{matrix}a − 1, 1 − a + c, b + 1, 1 − b + c, (A) \\ (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C59[m1,m2,m3,m4].

m1, m2, m3, m4 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C60 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, b, c − b, (A) \\ (B) \end{matrix} ; z] ⟶ \frac{a (c − a)}{(a − b) (c − a − b)}_{r}F_{s}[ \begin{matrix}a + 1, 1 − a + c, b, c − b, (A) \\ (B) \end{matrix} ; z]

− \frac{b (c − b)}{(a − b) (c − a − b)}_{r}F_{s}[ \begin{matrix}a, c − a, b + 1, 1 − b + c, (A) \\ (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C60[m1,m2,m3,m4].

m1, m2, m3, m4 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C61 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, b, c − b, (A) \\ (B) \end{matrix} ; z] ⟶ \frac{ab}{(c − a − 1) (c − b − 1)}_{r}F_{s}[ \begin{matrix}a + 1, c − a − 1, b + 1, −1 − b + c, (A) \\ (B) \end{matrix} ; z]

+ \begin{matrix}(c − 1) & (−1 − a − b + c) \\ (c − a − 1) & (c − b − 1) \end{matrix}_{r+1}F_{s+1}[ \begin{matrix}1 + \frac{c−1}{2}, a, −1 − a + c, b, c − b − 1, (A) \\ \frac{c−1}{2}, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C61[m1,m2,m3,m4].

m1, m2, m3, m4 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C62 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, b, 2 − b + c, (A) \\ (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a + 1, 1 − a + c, b − 1, 1 + c − b, (A) \\ (B) \end{matrix} ; z]

+ (a − b + 1) (1 + c) (c − a − b + 1) z \frac{∏_{i=1}^{r−4} A_{i}}{∏_{i=1}^{s} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a + 1, 1 − a + c, b, 2 − b + c, c + 2, (A + 1) \\ c + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C62[m1,m2,m3,m4].

m1, m2, m3, m4 are the positions of the special upper parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, 2 − a + c, b, c − b, (A) \\ (B) \end{matrix} ; z]

⟶ \frac{b (c − b) (a − d − 1) (1 − a + c − d)}{(a − 1) (1 − a + c) (b − d) (c − b − d)}_{r}F_{s}[ \begin{matrix}a − 1, 1 − a + c, b + 1, 1 − b + c, (A) \\ (B) \end{matrix} ; z]

+ \frac{(1 − a + b) (1 − a − b + c) (c − d) d}{(a − 1) (1 − a + c) (b − d) (c − b − d)}_{r+2}F_{s+2}[ \begin{matrix}a − 1, 1 − a + c, b, −b + c, d + 1, 1 + c − d, (A) \\ d, c − d, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C63[m1,m2,m3,m4,d].

m1, m2, m3, m4 are the positions of the special upper parameters, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

C64 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, 1 − a + c, b, (A) \\ c − b, (B) \end{matrix} ; z] ⟶ \begin{matrix}(a − b − 1) & (c − a − b) \\ (a − 1) & (c − a) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, c − a, b, (A) \\ c − b, (B) \end{matrix} ; z]

+ \frac{b (c − b − 1)}{(a − 1) (c − a)}_{r}F_{s}[ \begin{matrix}a − 1, c − a, b + 1, (A) \\ −1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C64[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively. \underline{Example(s)}:

In[1]:= F[{A,B,1+C,D},{E,A+C-D,F},z]
             A, B, 1 + C, D
Out[1]=  F                  ; z
        4 3  E, A + C - D, F
In[2]:= %/.C64[3,1,4,2]

             C, -1 + A, D, B
         F                  ; z   (-1 + A - D)  (C - D)
        4 3  A + C - D, E, F                  1        1
Out[2]= ------------------------------------------------ +
                         (-1 + A)  (C)
                                 1    1

          C, -1 + A, 1 + D, B
      F                       ; z   (-1 + A + C - D)  (D)
     4 3  -1 + A + C - D, E, F                      1    1
⟩  -----------------------------------------------------
                        (-1 + A)  (C)
                                1    1
In[3]:= %/.paufl

                                  C, -1 + A, D, B
        (-1 + A - D) (C - D)  F                  ; z
                             4 3  A + C - D, E, F
Out[3]= ---------------------------------------------- +
                          (-1 + A) C

                             C, -1 + A, 1 + D, B
     (-1 + A + C - D) D  F                       ; z
                        4 3  -1 + A + C - D, E, F
⟩  -------------------------------------------------
                        (-1 + A) C

The third, first, and fourth upper parameters in Out[1] are 1+C, A, and D, respectively, the second lower parameter in Out[1] is A+C-D. Hence C64 can be applied with the replacements a → 1+C, b → D and c → A+C.

\underline{See also}: ContigListe, Ers, PosListe.

C65

\underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, b, (A) \\ 1 − b + c, (B) \end{matrix} ; z] ⟶ \frac{a (c − a)}{(b − 1) (1 − b + c)}_{r}F_{s}[ \begin{matrix}a + 1, 1 − a + c, b − 1, (A) \\ 2 − b + c, (B) \end{matrix} ; z]

− \begin{matrix}(1 + a − b) & (1 − a − b + c) \\ (b − 1) & (1 − b + c) \end{matrix}_{r}F_{s}[ \begin{matrix}a, c − a, b − 1, (A) \\ 2 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C65[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C66 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, b, (A) \\ 1 − b + c, (B) \end{matrix} ; z] ⟶ \frac{a (c − a)}{(a − b) (c − a − b)}_{r}F_{s}[ \begin{matrix}a + 1, 1 − a + c, b, (A) \\ 1 − b + c, (B) \end{matrix} ; z]

− \frac{b (c − b)}{(a − b) (c − a − b)}_{r}F_{s}[ \begin{matrix}a, c − a, b + 1, (A) \\ c − b, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C66[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C67 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, 1 − a + c, b, (A) \\ c − b, (B) \end{matrix} ; z] ⟶ \frac{ab}{(c − a) (c − b)}_{r}F_{s}[ \begin{matrix}a + 1, c − a, b + 1, (A) \\ 1 − b + c, (B) \end{matrix} ; z]

+ \frac{c (c − a − b)}{(c − a) (c − b)}_{r+1}F_{s+1}[ \begin{matrix}1 + \frac{c}{2}, a, c − a, b, (A) \\ \frac{c}{2}, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C67[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C68 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, b, (A) \\ 1 − b + c, (B) \end{matrix} ; z] ⟶ \begin{matrix}(c − a) & (c − b) \\ (a − 1) & (b − 1) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, 1 − a + c, b − 1, (A) \\ c − b, (B) \end{matrix} ; z]

− \begin{matrix}(c − 1) & (1 − a − b + c) \\ (a − 1) & (b − 1) \end{matrix}_{r+1}F_{s+1}[ \begin{matrix}1 + \frac{c−1}{2}, −1 + a, c − a, b − 1, (A) \\ \frac{c−1}{2}, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C68[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, c − a, b, (A) \\ 1 − b + c, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a + 1, 1 − a + c, b − 1, (A) \\ 2 − b + c, (B) \end{matrix} ; z]

+ \frac{(1 + a − b) (1 + c) (1 − a − b + c) z}{(1 − b + c) (2 − b + c)} \frac{∏_{i=1}^{r−3} A_{i}}{∏_{i=1}^{s−1} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a + 1, 1 − a + c, b, c + 2, (A + 1) \\ 3 − b + c, c + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C69[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C70 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, 1 − a + c, b, (A) \\ c − b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a − 1, c − a, b + 1, (A) \\ −1 − b + c, (B) \end{matrix} ; z]

− \frac{(a − b − 1) c (c − a − b) z}{(c − b − 1) (c − b)} \frac{∏_{i=1}^{r−3} A_{i}}{∏_{i=1}^{s−1} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a, 1 − a + c, b + 1, c + 1, (A + 1) \\ 1 − b + c, c, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C70[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C71 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, 1 − a + c, b, (A) \\ c − b, (B) \end{matrix} ; z] ⟶ \frac{b (c − b − 1) (a − d − 1) (c − a − d)}{(a − 1) (c − a) (b − d) (c − b − 1 − d)}_{r}F_{s}[ \begin{matrix}a − 1, c − a, b + 1, (A) \\ −1 − b + c, (B) \end{matrix} ; z]

+ \frac{(1 − a + b) (c − a − b) (c − d − 1) d}{(a − 1) (c − a) (b − d) (c − b − 1 − d)}_{r+2}F_{s+2}[ \begin{matrix}a − 1, c − a, b, d + 1, c − d, (A) \\ c − b, d, c − d − 1, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C71[m1,m2,m3,n1,d].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, c − a, b, (A) \\ 1 − b + c, (B) \end{matrix} ; z] ⟶ \frac{a (c − a) (b − d − 1) (1 − b + c − d)}{(b − 1) (1 − b + c) (a − d) (c − a − d)}_{r}F_{s}[ \begin{matrix}a + 1, 1 − a + c, b − 1, (A) \\ 2 − b + c, (B) \end{matrix} ; z]

− \frac{(b − a − 1) (1 − a − b + c) (c − d) d}{(b − 1) (1 − b + c) (a − d) (c − a − d)}_{r+2}F_{s+2}[ \begin{matrix}a, c − a, b − 1, d + 1, 1 + c − d, (A) \\ 2 − b + c, d, c − d, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C72[m1,m2,m3,n1,d].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

C73 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, a + b, (A) \\ b − a + 1, (B) \end{matrix} ; z] ⟶ \frac{1}{(a + b − 1) z} \frac{∏_{i=1}^{s−1}(B_{i} − 1)}{∏_{i=1}^{r−3}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a, b, (A − 1) \\ (B − 1) \end{matrix} ; z]

− \frac{1}{(a + b − 1) z} \frac{∏_{i=1}^{s−1}(B_{i} − 1)}{∏_{i=1}^{r−3}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a − 1, b − 1, (A − 1) \\ (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C73[m1,m2,m3,n1].

m1, m2, m3 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C74 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, (A) \\ c − a, c − b, (B) \end{matrix} ; z] ⟶ \begin{matrix}(a − b − 1) & (c − a − b) \\ (a − 1) & (c − a) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, b, (A) \\ 1 − a + c, −b + c, (B) \end{matrix} ; z]

+ \frac{b (c − b − 1)}{(a − 1) (c − a)}_{r}F_{s}[ \begin{matrix}a − 1, b + 1, (A) \\ 1 − a + c, c − b − 1, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C74[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

Alphabetic List: C78 31

C75 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, (A) \\ c − a, c − b, (B) \end{matrix} ; z] ⟶ \frac{a (c − a − 1)}{(a − b) (−1 − a − b + c)}_{r}F_{s}[ \begin{matrix}a + 1, b, (A) \\ c − a − 1, −b + c, (B) \end{matrix} ; z]

− \frac{b (c − b − 1)}{(a − b) (−1 − a − b + c)}_{r}F_{s}[ \begin{matrix}a, b + 1, (A) \\ c − a, −1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C75[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C76 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, (A) \\ c − a, c − b, (B) \end{matrix} ; z] ⟶ \frac{ab}{(c − a) (c − b)}_{r}F_{s}[ \begin{matrix}a + 1, b + 1, (A) \\ 1 − a + c, 1 − b + c, (B) \end{matrix} ; z]

+ \frac{c (c − a − b)}{(c − a) (c − b)}_{r+1}F_{s+1}[ \begin{matrix}1 + \frac{c}{2}, a, b, (A) \\ \frac{c}{2}, 1 − a + c, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C76[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C77 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, (A) \\ c − a, c − b, (B) \end{matrix} ; z] ⟶ \begin{matrix}(c − a − 1) & (c − b − 1) \\ (a − 1) & (b − 1) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, b − 1, (A) \\ c − a − 1, c − b − 1, (B) \end{matrix} ; z]

− \begin{matrix}(c − 2) & (c − a − b) \\ (a − 1) & (b − 1) \end{matrix}_{r+1}F_{s+1}[ \begin{matrix}1 + \frac{c−2}{2}, −1 + a, b − 1, (A) \\ \frac{c−2}{2}, c − a, c − b, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C77[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C78 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, (A) \\ c − a, c − b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a + 1, b − 1, (A) \\ c − a − 1, 1 − b + c, (B) \end{matrix} ; z]

+ \frac{(1 + a − b) c (c − a − b) z}{(c − a − 1) (c − a) (c − b) (1 − b + c)} \frac{∏_{i=1}^{r−2} A_{i}}{∏_{i=1}^{s−2} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a + 1, b, c + 1, (A + 1) \\ 1 − a + c, 2 − b + c, c, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C78[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C79

\underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, (A) \\ c − a, c − b, (B) \end{matrix} ; z] ⟶ \frac{b (c − b − 1) (a − d − 1) (c − a − d)}{(a − 1) (c − a) (b − d) (c − b − 1 − d)}_{r}F_{s}[ \begin{matrix}a − 1, b + 1, (A) \\ 1 − a + c, c − b − 1, (B) \end{matrix} ; z]

+ \frac{(1 − a + b) (c − a − b) (c − d − 1) d}{(a − 1) (c − a) (b − d) (c − b − 1 − d)}_{r+2}F_{s+2}[ \begin{matrix}a − 1, b, d + 1, c − d, (A) \\ 1 − a + c, c − b, d, c − d − 1, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C79[m1,m2,n1,n2,d].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{Example(s)}:

In[1]:= F[{A,B,B+F-E,D},{E,F},z]

             A, B, B - E + F, D
Out[1]=  F                     ; z
        4 2         E, F

In[2]:= %/.C79[2,3,2,1,x]
Out[2]= ((-1 + E) (B - E + F) (-1 + B - x) (F - x)

             -1 + B, 1 + B - E + F, A, D
⟩  F                              ; z  ) /
        4 2         1 + F, -1 + E

⟩  ((-1 + B) F (-1 + E - x) (B - E + F - x)) +
⟩  ((-B + E) (1 - E + F) (-1 + B + F - x) x

             -1 + B, B - E + F, 1 + x, B + F - x, A, D
⟩  F                                            ; z  ) /
        6 4         1 + F, E, x, -1 + B + F - x

⟩  ((-1 + B) F (-1 + E - x) (B - E + F - x))

The second and third upper parameters in Out[1] are B, and B+F-E, respectively, the second and first lower parameters in Out[1] are F and E, respectively. Hence C79 can be applied with the replacements a → B, b → B+F-E and c → B+F. The x replaces d in the right-hand side expression.

\underline{See also}: C79, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, c − a, (A) \\ b, c − b, (B) \end{matrix} ; z] ⟶ \begin{matrix}(b − 1) & (c − b − 1) \\ (a − 1) & (c − a − 1) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, c − a − 1, (A) \\ b − 1, c − b − 1, (B) \end{matrix} ; z]

+ \begin{matrix}(a − b) & (c − a − b) \\ (a − 1) & (c − a − 1) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, c − a − 1, (A) \\ b, −b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C80[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C81 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, (A) \\ b, c − b, (B) \end{matrix} ; z] ⟶ \frac{a (c − a)}{b (c − b)}_{r}F_{s}[ \begin{matrix}a + 1, 1 − a + c, (A) \\ b + 1, 1 − b + c, (B) \end{matrix} ; z] − \frac{(a − b) (c − a − b)}{b (c − b)}_{r}F_{s}[ \begin{matrix}a, c − a, (A) \\ b + 1, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C81[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C82 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, (A) \\ b, 2 − b + c, (B) \end{matrix} ; z] ⟶ \frac{a (c − a)}{(1 + a − b) (1 − a − b + c)}_{r}F_{s}[ \begin{matrix}a + 1, 1 − a + c, (A) \\ b, 2 − b + c, (B) \end{matrix} ; z]

− \begin{matrix}(b − 1) & (1 − b + c) \\ (1 + a − b) & (1 − a − b + c) \end{matrix}_{r}F_{s}[ \begin{matrix}a, c − a, (A) \\ b − 1, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C82[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C83 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, (A) \\ b, c − b, (B) \end{matrix} ; z] ⟶ \frac{a (b − 1)}{(c − a − 1) (c − b)}_{r}F_{s}[ \begin{matrix}a + 1, −1 − a + c, (A) \\ b − 1, 1 − b + c, (B) \end{matrix} ; z]

+ \begin{matrix}(c − 1) & (c − a − b) \\ (c − a − 1) & (c − b) \end{matrix}_{r+1}F_{s+1}[ \begin{matrix}1 + \frac{c−1}{2}, a, −1 − a + c, (A) \\ \frac{c−1}{2}, b, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C83[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, c − a, (A) \\ b, c − b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a + 1, 1 − a + c, (A) \\ b + 1, 1 − b + c, (B) \end{matrix} ; z]

+ \frac{(a − b) (1 + c) (c − a − b) z}{b (1 + b) (c − b) (1 − b + c)} \frac{∏_{i=1}^{r−2} A_{i}}{∏_{i=1}^{s−2} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a + 1, 1 − a + c, c + 2, (A + 1) \\ b + 2, 2 − b + c, c + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C84[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C85 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, (A) \\ b, c − b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a − 1, c − a − 1, (A) \\ b − 1, −1 − b + c, (B) \end{matrix} ; z]

− \frac{(a − b) (c − 1) (c − a − b) z}{(b − 1) b (c − b − 1) (c − b)} \frac{∏_{i=1}^{r−2} A_{i}}{∏_{i=1}^{s−2} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a, c − a, c, (A + 1) \\ 1 + b, 1 − b + c, c − 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C85[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C86 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, (A) \\ b, c − b, (B) \end{matrix} ; z] ⟶ \begin{matrix}(b − 1) & (c − b − 1) & (a − d − 1) & (c − a − 1 − d) \\ (a − 1) & (c − a − 1) & (b − d − 1) & (c − b − 1 − d) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, c − a − 1, (A) \\ b − 1, c − b − 1, (B) \end{matrix} ; z]

+ \frac{(b − a) (c − a − b) (c − d − 2) d}{(a − 1) (c − a − 1) (b − d − 1) (c − b − 1 − d)}_{r+2}F_{s+2}[ \begin{matrix}a − 1, c − a − 1, d + 1, c − d − 1, (A) \\ b, c − b, d, c − d − 2, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C86[m1,m2,n1,n2,d].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, c − a, (A) \\ b, c − b, (B) \end{matrix} ; z] ⟶ \frac{a (c − a) (b − d) (c − b − d)}{b (c − b) (a − d) (c − a − d)}_{r}F_{s}[ \begin{matrix}a + 1, 1 − a + c, (A) \\ b + 1, 1 − b + c, (B) \end{matrix} ; z]

− \frac{(b − a) (c − a − b) (c − d) d}{b (c − b) (a − d) (c − a − d)}_{r+2}F_{s+2}[ \begin{matrix}a, c − a, d + 1, 1 + c − d, (A) \\ b + 1, 1 − b + c, d, c − d, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C87[m1,m2,n1,n2,d].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

C88 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, a + b, (A) \\ b + 1, b − a + 1, (B) \end{matrix} ; z] ⟶ \frac{(b − 1) b}{(a + b − 1) z} \frac{∏_{i=1}^{s−2}(B_{i} − 1)}{∏_{i=1}^{r−2}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a, (A − 1) \\ b − 1, (B − 1) \end{matrix} ; z]

− \frac{(b − 1) b}{(a + b − 1) z} \frac{∏_{i=1}^{s−2}(B_{i} − 1)}{∏_{i=1}^{r−2}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a − 1, (A − 1) \\ b, (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C88[m1,m2,n1,n2].

m1, m2 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C89 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ c − a, b, 1 − b + c, (B) \end{matrix} ; z] ⟶ \begin{matrix}(b − 1) & (c − b) \\ (a − 1) & (c − a) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, (A) \\ 1 − a + c, b − 1, c − b, (B) \end{matrix} ; z]

+ \begin{matrix}(a − b) & (1 − a − b + c) \\ (a − 1) & (c − a) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, (A) \\ 1 − a + c, b, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C89[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, (A) \\ 1 − a + c, b, c − b, (B) \end{matrix} ; z] ⟶ \frac{a (c − a)}{b (c − b)}_{r}F_{s}[ \begin{matrix}a + 1, (A) \\ c − a, b + 1, 1 − b + c, (B) \end{matrix} ; z]

− \frac{(a − b) (c − a − b)}{b (c − b)}_{r}F_{s}[ \begin{matrix}a, (A) \\ 1 − a + c, b + 1, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C90[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C91 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ c − a, b, 1 − b + c, (B) \end{matrix} ; z] ⟶ \frac{a (c − a − 1)}{(1 + a − b) (c − a − b)}_{r}F_{s}[ \begin{matrix}a + 1, (A) \\ c − a − 1, b, 1 − b + c, (B) \end{matrix} ; z]

− \begin{matrix}(b − 1) & (c − b) \\ (1 + a − b) & (c − a − b) \end{matrix}_{r}F_{s}[ \begin{matrix}a, (A) \\ c − a, b − 1, c − b, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C91[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C92 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ c − a, b, 1 − b + c, (B) \end{matrix} ; z] ⟶ \frac{a (b − 1)}{(c − a) (1 − b + c)}_{r}F_{s}[ \begin{matrix}a + 1, (A) \\ 1 − a + c, b − 1, 2 − b + c, (B) \end{matrix} ; z]

+ \frac{c (1 − a − b + c)}{(c − a) (1 − b + c)}_{r+1}F_{s+1}[ \begin{matrix}1 + \frac{c}{2}, a, (A) \\ \frac{c}{2}, 1 − a + c, b, 2 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C92[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C93 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ 1 − a + c, b, c − b, (B) \end{matrix} ; z] ⟶ \frac{(c − a) (c − b − 1)}{(a − 1) b}_{r}F_{s}[ \begin{matrix}a − 1, (A) \\ c − a, b + 1, −1 − b + c, (B) \end{matrix} ; z]

− \frac{(c − 1) (c − a − b)}{(a − 1) b}_{r+1}F_{s+1}[ \begin{matrix}1 + \frac{c−1}{2}, −1 + a, (A) \\ \frac{c−1}{2}, 1 − a + c, b + 1, c − b, (B) \end{matrix} ; z] ⟶

\underline{Usage}: Expr/.C93[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, (A) \\ 1 − a + c, b, c − b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a + 1, (A) \\ c − a, b + 1, 1 − b + c, (B) \end{matrix} ; z]

+ \frac{(a − b) (1 + c) (c − a − b) z}{b (1 + b) (c − a) (1 − a + c) (c − b) (1 − b + c)} \frac{∏_{i=1}^{r−1} A_{i}}{∏_{i=1}^{s−3} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a + 1, c + 2, (A + 1) \\ 2 − a + c, b + 2, 2 − b + c, c + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C94[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C95 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ c − a, b, 1 − b + c, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}a − 1, (A) \\ 1 − a + c, b − 1, −b + c, (B) \end{matrix} ; z]

− \frac{(a − b) c (1 − a − b + c) z}{(b − 1) b (c − a) (1 − a + c) (c − b) (1 − b + c)} \frac{∏_{i=1}^{r−1} A_{i}}{∏_{i=1}^{s−3} B_{i}}_{r+1}F_{s+1}[ \begin{matrix}a, c + 1, (A + 1) \\ 2 − a + c, b + 1, 2 − b + c, c, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C95[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C96 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, (A) \\ c − a, b, 1 − b + c, (B) \end{matrix} ; z] ⟶ \begin{matrix}(b − 1) & (c − b) & (a − d − 1) & (c − a − d) \\ (a − 1) & (c − a) & (b − d − 1) & (c − b − d) \end{matrix}_{r}F_{s}[ \begin{matrix}a − 1, (A) \\ 1 − a + c, b − 1, c − b, (B) \end{matrix} ; z]

+ \frac{(b − a) (1 − a − b + c) (c − d − 1) d}{(a − 1) (c − a) (b − d − 1) (c − b − d)}_{r+2}F_{s+2}[ \begin{matrix}a − 1, d + 1, c − d, (A) \\ 1 − a + c, b, 1 − b + c, d, c − d − 1, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C96[m1,n1,n2,n3,d].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, (A) \\ 1 − a + c, b, c − b, (B) \end{matrix} ; z] ⟶ \frac{a (c − a) (b − d) (c − b − d)}{b (c − b) (a − d) (c − a − d)}_{r}F_{s}[ \begin{matrix}a + 1, (A) \\ c − a, b + 1, 1 − b + c, (B) \end{matrix} ; z]

− \frac{(b − a) (c − a − b) (c − d) d}{b (c − b) (a − d) (c − a − d)}_{r+2}F_{s+2}[ \begin{matrix}a, d + 1, 1 + c − d, (A) \\ 1 − a + c, b + 1, 1 − b + c, d, c − d, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C97[m1,n1,n2,n3,d].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

C98 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a + b, (A) \\ a + 1, b + 1, b − a + 1, (B) \end{matrix} ; z] ⟶ \frac{(a − 1) a (b − 1) b}{(a + b − 1) z} \frac{∏_{i=1}^{s−3}(B_{i} − 1)}{∏_{i=1}^{r−1}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}(A − 1) \\ a − 1, b − 1, (B − 1) \end{matrix} ; z]

− \frac{(a − 1) a (b − 1) b}{(a + b − 1) z} \frac{∏_{i=1}^{s−3}(B_{i} − 1)}{∏_{i=1}^{r−1}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}(A − 1) \\ a, b, (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C98[m1,n1,n2,n3].

m1 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C99 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ a, c − a, b, 2 − b + c, (B) \end{matrix} ; z] ⟶ \frac{(b − 1) (1 − b + c)}{a (c − a)}_{r}F_{s}[ \begin{matrix}(A) \\ a + 1, 1 − a + c, b − 1, 1 − b + c, (B) \end{matrix} ; z]

+ \frac{(1 + a − b) (1 − a − b + c)}{a (c − a)}_{r}F_{s}[ \begin{matrix}(A) \\ a + 1, 1 − a + c, b, 2 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C99[n1,n2,n3,n4].

n1, n2, n3, n4 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}(A) \\ a, c − a, b, c − b, (B) \end{matrix} ; z] ⟶ \begin{matrix}(a − 1) & (c − a − 1) \\ (a − b) & (c − a − b) \end{matrix}_{r}F_{s}[ \begin{matrix}(A) \\ a − 1, c − a − 1, b, −b + c, (B) \end{matrix} ; z]

− \begin{matrix}(b − 1) & (c − b − 1) \\ (a − b) & (c − a − b) \end{matrix}_{r}F_{s}[ \begin{matrix}(A) \\ a, c − a, b − 1, −1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C100[n1,n2,n3,n4].

n1, n2, n3, n4 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C101 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ a, c − a, b, c − b, (B) \end{matrix} ; z] ⟶ \begin{matrix}(a − 1) & (b − 1) \\ (c − a) & (c − b) \end{matrix}_{r}F_{s}[ \begin{matrix}(A) \\ a − 1, 1 − a + c, b − 1, 1 − b + c, (B) \end{matrix} ; z]

+ \begin{matrix}(c − 1) & (1 − a − b + c) \\ (c − a) & (c − b) \end{matrix}_{r+1}F_{s+1}[ \begin{matrix}1 + \frac{c−1}{2}, (A) \\ \frac{c−1}{2}, a, 1 − a + c, b, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C101[n1,n2,n3,n4].

n1, n2, n3, n4 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C102 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}(A) \\ a, 2 − a + c, b, c − b, (B) \end{matrix} ; z] ⟶ _{r}F_{s}[ \begin{matrix}(A) \\ a − 1, 1 − a + c, b + 1, 1 − b + c, (B) \end{matrix} ; z]

+ \frac{(a − b − 1) (1 + c) (1 − a − b + c) z}{(a − 1) ab (1 + b) (1 − a + c) (2 − a + c) (c − b) (1 − b + c)} \frac{∏_{i=1}^{r} A_{i}}{∏_{i=1}^{s−4} B_{i}}

×_{r+1}F_{s+1}[ \begin{matrix}c + 2, (A + 1) \\ a + 1, 3 − a + c, b + 2, 2 − b + c, c + 1, (B + 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C102[n1,n2,n3,n4].

n1, n2, n3, n4 are the positions of the special lower parameters.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}(A) \\ a, c − a, b, 2 − b + c, (B) \end{matrix} ; z]

⟶ \frac{(b − 1) (1 − b + c) (a − d) (c − a − d)}{a (c − a) (b − d − 1) (1 − b + c − d)}_{r}F_{s}[ \begin{matrix}(A) \\ a + 1, 1 − a + c, b − 1, 1 − b + c, (B) \end{matrix} ; z]

+ \frac{(b − a − 1) (1 − a − b + c) (c − d) d}{a (c − a) (b − d − 1) (1 − b + c − d)}_{r+2}F_{s+2}[ \begin{matrix}d + 1, 1 + c − d, (A) \\ 1 + a, 1 − a + c, b, 2 − b + c, d, c − d, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C103[n1,n2,n3,n4,d].

n1, n2, n3, n4 are the positions of the special lower parameters, d is the additional parameter at the right hand side.

\underline{See also}: C79, ContigListe, Ers, PosListe.

C104 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, b, c − b, c, (A) \\ c − 1, (B) \end{matrix} ; z]

⟶ \frac{1}{(a − b) (c − 1) (c − a − b) z} \frac{∏_{i=1}^{s−1}(B_{i} − 1)}{∏_{i=1}^{r−5}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a − 1, c − a − 1, b, −b + c, (A − 1) \\ (B − 1) \end{matrix} ; z]

− \frac{1}{(a − b) (c − 1) (c − a − b) z} \frac{∏_{i=1}^{s−1}(B_{i} − 1)}{∏_{i=1}^{r−5}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a, c − a, b − 1, −1 − b + c, (A − 1) \\ (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C104[m1,m2,m3,m4,m5,n1].

m1, m2, m3, m4, m5 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C110 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}1 + \frac{c}{2}, a, c − a, b, c − b, (A) \\ \frac{c}{2}, (B) \end{matrix} ; z] ⟶ \frac{(c − a) (c − b)}{c (c − a − b)}_{r−1}F_{s−1}[ \begin{matrix}a, 1 − a + c, b, 1 − b + c, (A) \\ (B) \end{matrix} ; z]

− \frac{ab}{c (c − a − b)}_{r−1}F_{s−1}[ \begin{matrix}a + 1, c − a, b + 1, c − b, (A) \\ (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C110[m1,m2,m3,m4,m5,n1].

m1, m2, m3, m4, m5 and n1 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, c − a, b, c, (A) \\ 1 − b + c, −1 + c, (B) \end{matrix} ; z] ⟶ \frac{(c − b − 1) (c − b)}{(a − b) (c − 1) (c − a − b) z} \frac{∏_{i=1}^{s−2}(B_{i} − 1)}{∏_{i=1}^{r−4}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a − 1, c − a − 1, b, (A − 1) \\ c − b − 1, (B − 1) \end{matrix} ; z]

− \frac{(c − b − 1) (c − b)}{(a − b) (c − 1) (c − a − b) z} \frac{∏_{i=1}^{s−2}(B_{i} − 1)}{∏_{i=1}^{r−4}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a, c − a, b − 1, (A − 1) \\ c − b, (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C105[m1,m2,m3,m4,n1,n2].

m1, m2, m3, m4 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C112 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}1 + \frac{c}{2}, a, c − a, b, (A) \\ \frac{c}{2}, 1 − b + c, (B) \end{matrix} ; z] ⟶ \frac{(c − a) (c − b)}{c (c − a − b)}_{r−1}F_{s−1}[ \begin{matrix}a, 1 − a + c, b, (A) \\ c − b, (B) \end{matrix} ; z]

− \frac{ab}{c (c − a − b)}_{r−1}F_{s−1}[ \begin{matrix}a + 1, c − a, b + 1, (A) \\ 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C112[m1,m2,m3,m4,n1,n2].

m1, m2, m3, m4 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C106 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, c, (A) \\ 1 − a + c, 1 − b + c, −1 + c, (B) \end{matrix} ; z]

⟶ \frac{(c − a − 1) (c − a) (c − b − 1) (c − b)}{(a − b) (c − 1) (c − a − b) z} \frac{∏_{i=1}^{s−3}(B_{i} − 1)}{∏_{i=1}^{r−3}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a − 1, b, (A − 1) \\ −a + c, c − b − 1, (B − 1) \end{matrix} ; z]

− \frac{(c − a − 1) (c − a) (c − b − 1) (c − b)}{(a − b) (c − 1) (c − a − b) z} \frac{∏_{i=1}^{s−3}(B_{i} − 1)}{∏_{i=1}^{r−3}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a, b − 1, (A − 1) \\ −1 − a + c, c − b, (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C106[m1,m2,m3,n1,n2,n3].

m1, m2, m3 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, c − a, c, (A) \\ b + 1, 1 − b + c, −1 + c, (B) \end{matrix} ; z]

⟶ \frac{(b − 1) b (c − b − 1) (c − b)}{(a − b) (c − 1) (c − a − b) z} \frac{∏_{i=1}^{s−3}(B_{i} − 1)}{∏_{i=1}^{r−3}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}−1 + a, c − a − 1, (A − 1) \\ b − 1, c − b − 1, (B − 1) \end{matrix} ; z]

− \frac{(b − 1) b (c − b − 1) (c − b)}{(a − b) (c − 1) (c − a − b) z} \frac{∏_{i=1}^{s−3}(B_{i} − 1)}{∏_{i=1}^{r−3}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a, c − a, (A − 1) \\ b, c − b, (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C107[m1,m2,m3,n1,n2,n3].

m1, m2, m3 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C114 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}1 + \frac{c}{2}, a, b, (A) \\ \frac{c}{2}, 1 − a + c, 1 − b + c, (B) \end{matrix} ; z] ⟶ \frac{(c − a) (c − b)}{c (c − a − b)}_{r−1}F_{s−1}[ \begin{matrix}a, b, (A) \\ c − a, −b + c, (B) \end{matrix} ; z]

− \frac{ab}{c (c − a − b)}_{r−1}F_{s−1}[ \begin{matrix}a + 1, b + 1, (A) \\ 1 − a + c, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C114[m1,m2,m3,n1,n2,n3].

m1, m2, m3 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C115 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}1 + \frac{c}{2}, a, c − a, (A) \\ \frac{c}{2}, b, 2 − b + c, (B) \end{matrix} ; z] ⟶ \frac{(c − a) (1 − b + c)}{c (1 − a − b + c)}_{r−1}F_{s−1}[ \begin{matrix}a, 1 − a + c, (A) \\ b, 1 − b + c, (B) \end{matrix} ; z]

− \frac{a (b − 1)}{c (1 − a − b + c)}_{r−1}F_{s−1}[ \begin{matrix}a + 1, c − a, (A) \\ −1 + b, 2 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C115[m1,m2,m3,n1,n2,n3].

m1, m2, m3 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

rF_{s}[ \begin{matrix}a, c, (A) \\ 1 − a + c, b, 2 − b + c, −1 + c, (B) \end{matrix} ; z]

⟶ \frac{(b − 2) (b − 1) (c − a − 1) (c − a) (c − b) (1 − b + c)}{(1 + a − b) (c − 1) (1 − a − b + c) z} \frac{∏_{i=1}^{s−4}(B_{i} − 1)}{∏_{i=1}^{r−2}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a − 1, (A − 1) \\ c − a, b − 2, c − b, (B − 1) \end{matrix} ; z]

− \frac{(b − 2) (b − 1) (c − a − 1) (c − a) (c − b) (1 − b + c)}{(1 + a − b) (c − 1) (1 − a − b + c) z} \frac{∏_{i=1}^{s−4}(B_{i} − 1)}{∏_{i=1}^{r−2}(A_{i} − 1)}_{r−1}F_{s−1}[ \begin{matrix}a, (A − 1) \\ c − a − 1, b − 1, 1 − b + c, (B − 1) \end{matrix} ; z]

\underline{Usage}: Expr/.C108[m1,m2,n1,n2,n3,n4].

m1, m2 and n1, n2, n3, n4 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C118 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}1 + \frac{c}{2}, a, (A) \\ \frac{c}{2}, 1 − a + c, b, 2 − b + c, (B) \end{matrix} ; z] ⟶ \frac{(c − a) (1 − b + c)}{c (1 − a − b + c)}_{r−1}F_{s−1}[ \begin{matrix}a, (A) \\ c − a, b, 1 − b + c, (B) \end{matrix} ; z]

− \frac{a (b − 1)}{c (1 − a − b + c)}_{r−1}F_{s−1}[ \begin{matrix}a + 1, (A) \\ 1 − a + c, b − 1, 2 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C118[m1,m2,n1,n2,n3,n4].

m1, m2 and n1, n2, n3, n4 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C109 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}c, (A) \\ a, 2 − a + c, b, 2 − b + c, −1 + c, (B) \end{matrix} ; z]

⟶ \frac{(a − 2) (a − 1) (b − 2) (b − 1) (c − a) (1 − a + c) (c − b) (1 − b + c)}{(a − b) (c − 1) (2 − a − b + c) z} \frac{∏_{i=1}^{s−5}(B_{i} − 1)}{∏_{i=1}^{r−1}(A_{i} − 1)}

×_{r−1}F_{s−1}[ \begin{matrix}(A − 1) \\ a − 1, 1 − a + c, b − 2, c − b, (B − 1) \end{matrix} ; z]

− \frac{(a − 2) (a − 1) (b − 2) (b − 1) (c − a) (1 − a + c) (c − b) (1 − b + c)}{(a − b) (c − 1) (2 − a − b + c) z} \frac{∏_{i=1}^{s−5}(B_{i} − 1)}{∏_{i=1}^{r−1}(A_{i} − 1)}

×_{r−1}F_{s−1}[ \begin{matrix}(A − 1) \\ a − 2, c − a, b − 1, 1 − b + c, (B − 1) \end{matrix} ; z]

44 Alphabetic List: C113

\underline{Usage}: Expr/.C109[m1,n1,n2,n3,n4,n5].

m1 and n1, n2, n3, n4, n5 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C120 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}1 + \frac{c}{2}, (A) \\ \frac{c}{2}, a, 2 − a + c, b, 2 − b + c, (B) \end{matrix} ; z] ⟶ \frac{(1 − a + c) (1 − b + c)}{c (2 − a − b + c)}_{r−1}F_{s−1}[ \begin{matrix}(A) \\ a, 1 − a + c, b, 1 − b + c, (B) \end{matrix} ; z]

− \frac{(a − 1) (b − 1)}{c (2 − a − b + c)}_{r−1}F_{s−1}[ \begin{matrix}(A) \\ a − 1, 2 − a + c, b − 1, 2 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C120[m1,n1,n2,n3,n4,n5].

m1 and n1, n2, n3, n4, n5 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C111 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, b, c − b, d + 1, 1 + c − d, (A) \\ d, c − d, (B) \end{matrix} ; z]

⟶ \frac{a (c − a) (b − d) (c − b − d)}{(b − a) (c − a − b) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}a + 1, 1 − a + c, b, −b + c, (A) \\ (B) \end{matrix} ; z]

− \frac{b (c − b) (a − d) (c − a − d)}{(b − a) (c − a − b) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}a, c − a, b + 1, 1 − b + c, (A) \\ (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C111[m1,m2,m3,m4,m5,m6,n1,n2].

m1, m2, m3, m4, m5, m6 and n1, n2 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C113 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, b, d + 1, 1 + c − d, (A) \\ 1 − b + c, d, c − d, (B) \end{matrix} ; z] ⟶ \frac{a (c − a) (b − d) (c − b − d)}{(b − a) (c − a − b) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}a + 1, 1 − a + c, b, (A) \\ 1 − b + c, (B) \end{matrix} ; z]

− \frac{b (c − b) (a − d) (c − a − d)}{(b − a) (c − a − b) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}a, c − a, b + 1, (A) \\ c − b, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C113[m1,m2,m3,m4,m5,n1,n2,n3].

m1, m2, m3, m4, m5 and n1, n2, n3 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

Alphabetic List: C121 45

C116 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, b, d + 1, 1 + c − d, (A) \\ 1 − a + c, 1 − b + c, d, c − d, (B) \end{matrix} ; z] ⟶ \frac{a (c − a) (b − d) (c − b − d)}{(b − a) (c − a − b) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}a + 1, b, (A) \\ c − a, 1 − b + c, (B) \end{matrix} ; z]

− \frac{b (c − b) (a − d) (c − a − d)}{(b − a) (c − a − b) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}a, b + 1, (A) \\ 1 − a + c, c − b, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C116[m1,m2,m3,m4,n1,n2,n3,n4].

m1, m2, m3, m4 and n1, n2, n3, n4 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C117 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, c − a, d + 1, 1 + c − d, (A) \\ b, 2 − b + c, d, c − d, (B) \end{matrix} ; z] ⟶ \frac{a (c − a) (b − d − 1) (1 − b + c − d)}{(b − a − 1) (1 − a − b + c) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}a + 1, 1 − a + c, (A) \\ b, 2 − b + c, (B) \end{matrix} ; z]

− \frac{(b − 1) (1 − b + c) (a − d) (c − a − d)}{(b − a − 1) (1 − a − b + c) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}a, c − a, (A) \\ −1 + b, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C117[m1,m2,m3,m4,n1,n2,n3,n4].

m1, m2, m3, m4 and n1, n2, n3, n4 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C119 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}a, d + 1, 1 + c − d, (A) \\ 1 − a + c, b, 2 − b + c, d, c − d, (B) \end{matrix} ; z]

⟶ \frac{a (c − a) (b − d − 1) (1 − b + c − d)}{(b − a − 1) (1 − a − b + c) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}a + 1, (A) \\ c − a, b, 2 − b + c, (B) \end{matrix} ; z]

− \frac{(b − 1) (1 − b + c) (a − d) (c − a − d)}{(b − a − 1) (1 − a − b + c) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}a, (A) \\ 1 − a + c, −1 + b, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C119[m1,m2,m3,n1,n2,n3,n4,n5].

m1, m2, m3 and n1, n2, n3, n4, n5 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

C121 \underline{Description}: Contiguous relation in form of a rule.

rF_{s}[ \begin{matrix}d + 1, 1 + c − d, (A) \\ a, 2 − a + c, b, 2 − b + c, d, c − d, (B) \end{matrix} ; z]

⟶ \frac{(a − 1) (1 − a + c) (b − d − 1) (1 − b + c − d)}{(b − a) (2 − a − b + c) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}(A) \\ a − 1, 1 − a + c, b, 2 − b + c, (B) \end{matrix} ; z]

− \frac{(b − 1) (1 − b + c) (a − d − 1) (1 − a + c − d)}{(b − a) (2 − a − b + c) (c − d) d}_{r−2}F_{s−2}[ \begin{matrix}(A) \\ a, 2 − a + c, −1 + b, 1 − b + c, (B) \end{matrix} ; z]

\underline{Usage}: Expr/.C121[m1,m2,n1,n2,n3,n4,n5,n6].

m1, m2 and n1, n2, n3, n4, n5, n6 are the positions of the special upper and lower parameters, respectively.

\underline{See also}: C64, ContigListe, Ers, PosListe.

ContigListe

\underline{Description}: List of all contiguous relations.

\underline{Usage}: ContigListe.

Div

\underline{Description}: Function that divides Gleichung by Expr.

\underline{Usage}: Div[Expr].

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
                           (-a + c)
             a, -n                 n
Out[1]=  F        ; 1   == ---------
        2 1    c             (c)
                                n
In[2]:= Div[a^{∧}n]

             a, -n
         F        ; 1
        2 1    c           (-a + c)
                                   n
Out[2]= --------------- == ---------
               n             n
              a             a  (c)
                                  n
In[3]:= Gleichung

             a, -n
         F        ; 1
        2 1    c           (-a + c)
                                   n
Out[3]= --------------- == ---------
               n             n
              a             a  (c)
                                  n
\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Mal, Add, Sub, Hoch, GlTausche, Ers.

Drucke

\underline{Description}: Function that directly sends an expression Expr in the Form PrintedForm to the printer. This function only works for DOS-machines with a printer directly connected. PrintedForm is an optional parameter which can be any of the format types (InputForm, OutputForm, TeXForm, …). The default is OutputForm.

\underline{Usage}: Drucke[Expr,PrintedForm].
\underline{See also}: TeXMat, AmSTeX, AmSLaTeX, LaTeX, TeX, TeXFV.

Ers

\underline{Description}: Function for controlled application of rules and functions.

\underline{Usage}: Ers[Expr,Rules,PosList].

Rules can be a rule, a list of rules, or a function. PosList must be a list of positions in Expr to which Rules should be applied. For instance, if PosList={{1, 2}, {4}}, then Rules is applied to Expr[[1,2]] and Expr[[4]] in Expr. If PosList={2, 3}, then Rules is applied to Expr[[2]] and Expr[[3]] in Expr. The positions of subexpressions can be determined by the function PosListe. If Ers is applied to an equation then the new left-hand and right-hand sides are automatically stored in the variables LS and RS.

There is an exceptional usage of Ers, namely

       Ers[Rules].

In this case the Rules are applied to both sides of the equation that is currently stored in Gleichung. Again, the new left-hand and right-hand sides are automatically stored in the variables LS and RS.

\underline{Example(s)}:

In[1]:= (-1)^{∧}n*p[a+1,n]*p[c,k]/p[-b,m]/p[1-d,l]
            n
        (-1)  (1 + a)  (c)
                     n    k
Out[1]= -------------------
          (-b)  (1 - d)
              m        l
In[2]:= PosListe[%]
              n                                1
Out[2]= {{(-1) , {{1}}}, {(1 + a) , {{2}}}, {-----, {{3}}}, {(c) , {{4}}},
                                 n           (-b)               k
                                                 m
         1
⟩  {--------, {{5}}}}
      (1 - d)
             l
In[3]:= Ers[%%,neg1,{5}]

            n
        (-1)  (1 + a)  (c)  (1 - d + l)
                     n    k            -l
Out[3]= ---------------------------------
                      (-b)
                          m
In[4]:= Ers[%%%,neg1,{2,4}]
                            n
                        (-1)
Out[4]= --------------------------------------
        (-b)  (1 - d)  (c + k)   (1 + a + n)
            m        l        -k            -n
In[5]:= Sum[%,{k,0,Infinity}]

∞

                                 n
         \  (-1)
Out[5]= ⟩ --------------------------------------
         /   (-b)  (1 - d)  (c + k)   (1 + a + n)
                 m        l        -k            -n
         k=0
In[6]:= PosListe[%,2]
                             n
Out[6]= {{0, {{2, 2}}}, {(-1) , {{1, 1}}}, {k, {{2, 1}}}, {∞, {{2, 3}}},
        1                   1                      1
⟩  {-----, {{1, 2}}}, {--------, {{1, 3}}}, {---------, {{1, 4}}},
      (-b)               (1 - d)               (c + k)
          m                     l                     -k
            1
⟩  {-------------, {{1, 5}}}}
      (1 + a + n)
                 -n
In[7]:= Ers[%%,neg1,{{1,3}}]

          ∞  n

                  (-1)  (1 - d + l)
         \  -l
Out[7]= ⟩ -----------------------------
         /   (-b)  (c + k)   (1 + a + n)
                 m        -k            -n
         k=0
In[8]:= Ers[%%%,neg1,{{1,4},{1,5}}]

          ∞  n

             (-1)  (1 + a)  (c)
         \  n    k
Out[8]= ⟩ -------------------
         /     (-b)  (1 - d)
                   m        l
         k=0
In[9]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
                           (-a + c)
             a, -n                 n
Out[9]=  F        ; 1   == ---------
        2 1    c             (c)
                                n
In[10]:= Ers[a-⟩1-a]
                                (-1 + a + c)
              1 - a, -n                     n
Out[10]=  F            ; 1   == -------------
         2 1      c                 (c)
                                       n
In[11]:= Gleichung
                                (-1 + a + c)
              1 - a, -n                     n
Out[11]=  F            ; 1   == -------------
         2 1      c                 (c)
                                       n
In[12]:= P
In[13]:= p[a,4]+1/p[b,3]

                                             1
Out[13]= a (1 + a) (2 + a) (3 + a) + -----------------
                                     b (1 + b) (2 + b)
In[14]:= Ers[%,Expand,{1}]
                   2      3    4           1
Out[14]= 6 a + 11 a  + 6 a  + a  + -----------------
                                   b (1 + b) (2 + b)
\underline{See also}: PosListe, ManipulationsListe, Subst.

erw1

\underline{Description}: \begin{matrix}(a)_{n} → (a)_{m+n}/(a + n)_{m}, \\ (a)_{n} → Γ(a + n)/Γ(a). \end{matrix}

The parameter m has to be entered on request. To apply the second rule, m has to be Infinity.

\underline{Usage}: Expr/.erw1.

\underline{Example(s)}:

In[1]:= p[a,n+1]
Out[1]= (a)
           1 + n
In[2]:= %/.erw1
top-extend by:  m-1
            (a)
               m + n
Out[2]= -----------------
        (1 + a + n)
                   -1 + m
In[3]:= p[-a+1,m]
Out[3]= (1 - a)
               m
In[4]:= %/.erw1
top-extend by:  Infinity
        Γ(1 - a + m)
Out[4]= ------------
          Γ(1 - a)
\underline{See also}: erw2, zus1, zus2, zus3, Ers, PosListe, ManipulationsListe.

erw2

\underline{Description}: \begin{matrix}(a)_{n} → (a − m)_{m+n}/(a − m)_{m}, \\ Γ(a) → Γ(a − m) · (a − m)_{m}. \end{matrix}

The parameter m has to be entered on request.

\underline{Usage}: Expr/.erw2.

\underline{Example(s)}:

In[1]:= p[a-1,n+1]
Out[1]= (-1 + a)
                1 + n
In[2]:= %/.erw2
bottom-extend by:  m-1
        (a - m)
               m + n
Out[2]= -------------
        (a - m)
               -1 + m
In[3]:= GAMMA[-b]
Out[3]= Γ(-b)
In[4]:= %/.erw2
bottom-extend by:  n
Out[4]= Γ(-b - n) (-b - n)
                          n
\underline{See also}: erw1, zus1, zus2, zus3, Ers, PosListe, ManipulationsListe.

Expandq

\underline{Description}: Rule that expands all the exponents in powers.

\underline{Usage}: Expr/.Expandq.

\underline{Example(s)}:

In[1]:= (-1)^{∧}m*p[a,m+1]
            m
Out[1]= (-1)  (a)
                 1 + m
In[2]:= %/.baszerl
split into ?  terms:  2

            m  1 + m  a            1 + a
Out[2]= (-1)  2      (-)          (-----)
                      2 (1 + m)/2    2   (1 + m)/2
In[3]:= Ers[%,trans,{3}]
            m + (1 + m)/2  1 + m  1 + a                a   1 + m
Out[3]= (-1)              2      (-----)          (1 - - - -----)
                                    2   (1 + m)/2      2     2   (1 + m)/2
In[4]:= %/.Expandq
            1/2 + (3 m)/2  1 + m  1 + a                a   1 + m
Out[4]= (-1)              2      (-----)          (1 - - - -----)
                                    2   (1 + m)/2      2     2   (1 + m)/2

\underline{See also}: SimplifyP, MinusOne, SumExpand, Ers, PosListe.

F

\underline{Description}: F[List1,List2,z] is the hypergeometric series with upper parameters List1, lower parameters List2, and argument z.

\underline{Usage}: F[List1,List2,z].

\underline{Example(s)}:

In[1]:= F[{a,b},{c},z]

             a, b
Out[1]=  F       ; z
        2 1   c

In[2]:= F[{a,b,c},{d,e,f},z]

             a, b, c
Out[2]=  F          ; z
        3 3  d, e, f

\underline{See also}:  SListe, TListe, SumRegeln, SumF, FSum, p, GAMMA, FCancel, FOrdne, FPerm, FTausche, P,
  FFormat.

Factorialp

\underline{Description}: Factorialp[n,k] is the usual factorial, written in terms of factorial symbols (Pochhammer symbols) p.

\underline{Usage}: Factorialp[n].

\underline{Example(s)}:

In[1]:= Factorialp[n]
Out[1]= (1)
           n
In[2]:= Factorialp[5]
Out[2]= (1)
           5
\underline{See also}: Binomialp.

FCancel

\underline{Description}: Switch that activates automatic cancelling of the upper and lower parameters in F[], or makes it inactive, respectively. By default automatic cancelling is active.

\underline{Usage}: FCancel.

\underline{Example(s)}:

In[1]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.
In[2]:= F[{a,b},{a,c},z]

             b
Out[2]=  F    ; z
        1 1  c

In[3]:= p[{a,b},{a,c},n]
        (b)
           n
Out[3]= ----
        (c)
           n
In[4]:= GAMMA[{a,b},{a,c}]

           b
Out[4]= Γ
           c

In[5]:= FCancel
In[6]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is inactive.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.
In[7]:= F[{a,b},{a,c},z]

             a, b
Out[7]=  F       ; z
        2 2  a, c

In[8]:= p[{a,b},{a,c},n]
        (b)
           n
Out[8]= ----
        (c)
           n
In[9]:= GAMMA[{a,b},{a,c}]

           b
Out[9]= Γ
           c

In[10]:= FCancel

In[11]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.

\underline{See also}: F, V, hypAttributes.

FEinf

\underline{Description}: Rule that inactivates automatic cancelling in F[] and then adds a parameter which has to be entered on request to the upper and lower parameters of F[].

\underline{Usage}: Expr/.FEinf.

\underline{Example(s)}:

In[1]:= F[{b,c,1},{1+a-c,1+a-b},z]

                   b, c, 1
Out[1]=  F                       ; z
        3 2  1 + a - c, 1 + a - b

In[2]:= %/.FEinf
Add the parameter:  a

                   a, b, c, 1
Out[2]=  F                          ; z
        4 3  a, 1 + a - c, 1 + a - b

\underline{See also}: FOrdne, FPerm, FTausche, PSort, SumRegeln, SumUmkehr, Ers, PosListe.

FFormat

\underline{Description}: Switch that activates hypergeometric output, or makes it inactive, respectively. By default hypergeometric output is active.

\underline{Usage}: FFormat.

\underline{Example(s)}:

In[1]:= p[a,n]/GAMMA[b/2]*F[{c,d},{c*d},z]

             c, d
         F       ; z   (a)
        2 1  c d          n
Out[1]= -------------------
                 b
               Γ(-)
                 2
In[2]:= Tgl5402
Do you want to set values for the equation?  [y|n]:  n

Format::toobig:  Expression too big for output.
                 Enter "FFormat" and retry.

In[3]:= FFormat
In[4]:= %%
Out[4]= F[{-n, b, c, d, e}, {1 - b - n, 1 - c - n, 1 - d - n,
⟩  -2 + 2 b + 2 c + 2 d + e + 2 n}, 1] ==
                             3   b   c   d
⟩  F[{1 - b - c - d - 2 n, - - - - - - - - n, 1 - c - d - n, 1 - b - d - n,
                             2   2   2   2
                       -n  1   n
⟩  1 - b - c - n, --, - - -, e, 3 - 2 b - 2 c - 2 d - e - 3 n},
                       2   2   2
        1   b   c   d
⟩  {- - - - - - - - n, 1 - b - n, 1 - c - n, 1 - d - n,
        2   2   2   2
                        3 n  3               3 n
⟩  2 - b - c - d - ---, - - b - c - d - ---, 2 - b - c - d - e - 2 n,
                         2   2                2
⟩  -1 + b + c + d + e + n}, 1]
⟩  p[{2 - b - c - d - e - 2 n, 3 - 2 b - 2 c - 2 d - 3 n},
⟩  {2 - b - c - d - 2 n, 3 - 2 b - 2 c - 2 d - e - 3 n}, n]

In[5]:= %1
        F[{c, d}, {c d}, z] p[a, n]
Out[5]= ---------------------------
                       b
                 GAMMA[-]
                       2
In[6]:= FFormat
In[7]:= %1

             c, d
         F       ; z   (a)
        2 1  c d          n
Out[7]= -------------------
                 b
               Γ(-)
                 2

FOrdne

\underline{Description}: Rule that tries to order the parameters of a hypergeometric series in “well-poised” order. If the parameters could be paired such that the sum of each pair equals A, however A is missing in the upper parameters, then you have to add A to the upper and lower parameters by FEinf before applying FOrdne.

\underline{Usage}: Expr/.FOrdne.

\underline{Example(s)}:

In[1]:= F[{-n,b,1+a/2,a},{a+1-b,a/2,a+1+n},z]
                            a
                 -n, b, 1 + -, a
                            2
Out[1]=  F                          ; z
        4 3             a
             1 + a - b, -, 1 + a + n
                        2
In[2]:= %/.FOrdne

                        a
                 a, 1 + -, -n, b
                        2
Out[2]=  F                          ; z
        4 3  a
             -, 1 + a + n, 1 + a - b
             2
In[3]:= F[{b,c,1},{a+1-c,a+1-b},z]

                   b, c, 1
Out[3]=  F                       ; z
        3 2  1 + a - c, 1 + a - b

In[4]:= %/.FOrdne

                   b, c, 1
Out[4]=  F                       ; z
        3 2  1 + a - c, 1 + a - b

In[5]:= %/.FEinf
Add the parameter:  a

                   a, b, c, 1
Out[5]=  F                          ; z
        4 3  a, 1 + a - c, 1 + a - b

In[6]:= %/.FOrdne

                   a, b, c, 1
Out[6]=  F                          ; z
        4 3  1 + a - b, 1 + a - c, a

In[7]:= FCancel
In[8]:= %%/.FOrdne

                   b, c, 1
Out[8]=  F                       ; z
        3 2  1 + a - b, 1 + a - c

\underline{See also}: FEinf, FPerm, FTausche, F, V, PSort, Ers, PosListe.

FPerm

\underline{Description}: Rule for permuting parameters in basic hypergeometric series.

\underline{Usage}: Expr/.FPerm[⟨Permutation⟩,x].

x can be u, l, b. u causes a permutation of upper parameters, l causes a permutation of lower parameters, b causes a simultaneous permutation of respective upper and lower parameters. Permutation must be a sequence of positive numbers forming a permutation. The effect is that the new parameter at position i is the old parameter from

  position Permutation[i].

\underline{Example(s)}:

In[1]:= F[{a,b,c,d},{e,f,g},z]

             a, b, c, d
Out[1]=  F             ; z
        4 3   e, f, g

In[2]:= %1/.FPerm[3,2,1,u]

             c, b, a, d
Out[2]=  F             ; z
        4 3   e, f, g

In[3]:= %1/.FPerm[3,2,1,l]

             a, b, c, d
Out[3]=  F             ; z
        4 3   g, f, e

In[4]:= %1/.FPerm[2,1,b]

             a, c, b, d
Out[4]=  F             ; z
        4 3   f, e, g

\underline{See also}: FTausche, FOrdne, F, V, PSort, Ers, PosListe.

FSum

\underline{Description}: Rule that transforms a F[] into a Sum[].

\underline{Usage}: Expr/.FSum.

\underline{Example(s)}:

In[1]:= F[{a,b},{c},z]

             a, b
Out[1]=  F       ; z
        2 1   c

In[2]:= %/.FSum
Is -a a nonnegative integer?
[y|n]:    n
Is -b a nonnegative integer?
[y|n]:    n
A hypergeometric series is converted into a sum.
Enter a variable for the summation index:  k

          ∞ k

             z  (a)  (b)
         \  k    k
Out[2]= ⟩ ------------
         /    (1)  (c)
                 k    k
         k=0
In[3]:= %1/.FSum
Is -a a nonnegative integer?
[y|n]:    n
Is -b a nonnegative integer?
[y|n]:    y
A hypergeometric series is converted into a sum.
Enter a variable for the summation index:  j

         -b   j
             z  (a)  (b)
         \  j    j
Out[3]= ⟩ ------------
         /    (1)  (c)
                 j    j
         j=0
\underline{See also}: F, SumF, Ers, PosListe.

FTausche

\underline{Description}: Rule for reordering parameters in hypergeometric series.

\underline{Usage}: Expr/.FTausche[n1,n2,x].

x can be u, l, b. u causes a reordering of upper parameters, l causes a reordering of lower parameters, b causes a simultaneous reordering of respective upper and lower parameters. n1 is the position of the parameter to be reordered, n2 is the new position.

\underline{Example(s)}:

In[1]:= F[{-n,b,1+a/2,a},{a+1-b,a/2,a+1+n},z]
                            a
                 -n, b, 1 + -, a
                            2
Out[1]=  F                          ; z
        4 3             a
             1 + a - b, -, 1 + a + n
                        2
In[2]:= %/.FTausche[1,4,u]
                        a
                 b, 1 + -, a, -n
                        2
Out[2]=  F                          ; z
        4 3             a
             1 + a - b, -, 1 + a + n
                        2
In[3]:= %/.FTausche[3,2,l]
                        a
                 b, 1 + -, a, -n
                        2
Out[3]=  F                          ; z
        4 3                        a
             1 + a - b, 1 + a + n, -

In[4]:= %/.FTausche[1,3,b]
                               a
                 b, a, -n, 1 + -
                               2
Out[4]=  F                          ; z
        4 3             a
             1 + a + n, -, 1 + a - b
                        2
\underline{See also}: FPerm, FOrdne, F, V, PSort, Ers, PosListe.

GAMMA

\underline{Description}: GAMMA[x] is the Gamma function Γ(x). GAMMA[List1,List2] is also provided as the usual abbreviation for the quotient of Gamma functions (cf. the example for Gzerl).

\underline{Usage}: GAMMA[x]
   or: GAMMA[List1,List2].

\underline{Example(s)}:

In[1]:= GAMMA[2*a+1]
Out[1]= Γ(1 + 2 a)
In[2]:= GAMMA[{a,b},{c,d}]

           a, b
Out[2]= Γ
           c, d

\underline{See also}: p, F, P, Gzerl, Gzus, FFormat.

Gleichung

\underline{Description}: Is a variable which stores equations. The equation Gleichung can be manipulated using the functions Add, Sub, Mal, Div, Hoch, GlTausche, Ers, and Sum[k,m,n], where m and n are integers or variables. The last command causes the equation to be summed over k from m to n. The parameter k is optional. It will be set kk, ii, jj, ll, mm, or nn, if it is omitted.

\underline{Usage}: Gleichung.

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n

                           (-a + c)
             a, -n                 n
Out[1]=  F        ; 1   == ---------
        2 1    c             (c)
                                n
In[2]:= Gleichung
                           (-a + c)
             a, -n                 n
Out[2]=  F        ; 1   == ---------
        2 1    c             (c)
                                n
In[3]:= Add[1]
                                   (-a + c)
                 a, -n                     n
Out[3]= 1 +  F        ; 1   == 1 + ---------
            2 1    c                 (c)
                                        n
In[4]:= Gleichung
                                   (-a + c)
                 a, -n                     n
Out[4]= 1 +  F        ; 1   == 1 + ---------
            2 1    c                 (c)
                                        n
In[5]:= LS=5
Out[5]= 5
In[6]:= Gleichung
                 (-a + c)
                         n
Out[6]= 5 == 1 + ---------
                   (c)
                      n
In[7]:= Sum[n,0,m]

          m         m        m
                                (-a + c)

\  \  \  n

Out[7]= ⟩ 5 == ⟩ 1 + ⟩ ---------
         /         /        /     (c)
                                     n
         n=0       n=0      n=0
In[8]:= Sum[0,M]
          M    m         M    m        M    m
                                               (-a + c)

\  \  \  \  \  \  n

Out[8]= ⟩  ⟩ 5 == ⟩  ⟩ 1 + ⟩  ⟩ ---------
         /    /         /    /        /    /     (c)
                                                    n
        kk=0  n=0      kk=0  n=0     kk=0  n=0
In[9]:= Gleichung
          M    m         M    m
                                     (-a + c)

\  \  \  \  n

Out[9]= ⟩  ⟩ 5 == ⟩  ⟩ 1 + ---------
         /    /         /    /         (c)
                                          n
        kk=0  n=0      kk=0  n=0
\underline{See also}:  SumListe$gl, TransListe$gl, LS, RS, Mal, Add, Div, Sub, Hoch, GlTausche, Ers, Subst,
  PSort.

GlTausche

\underline{Description}: GlTausche interchanges right-hand and left-hand sides in Gleichung.

\underline{Usage}: GlTausche.

\underline{Example(s)}:

In[1]:= Sgl3201
Do you want to set values for the equation?  [y|n]:  n
                                          (-a + c)  (-b + c)
                   a, b, -n                       n         n
Out[1]=  F                       ; 1   == -------------------
        3 2  c, 1 + a + b - c - n         (c)  (-a - b + c)
                                             n             n

                                          (-a + c)  (-b + c)
                   a, b, -n                       n         n
Out[2]=  F                       ; 1   == -------------------
        3 2  c, 1 + a + b - c - n         (c)  (-a - b + c)
                                             n             n
In[3]:= GlTausche
        (-a + c)  (-b + c)
                n         n               a, b, -n
Out[3]= ------------------- ==  F                       ; 1
        (c)  (-a - b + c)      3 2  c, 1 + a + b - c - n
           n             n
In[4]:= Gleichung
        (-a + c)  (-b + c)
                n         n               a, b, -n
Out[4]= ------------------- ==  F                       ; 1
        (c)  (-a - b + c)      3 2  c, 1 + a + b - c - n
           n             n

\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Mal, Add, Div, Sub, Hoch, Ers, Subst.

GOSPER

\underline{Description}: Rule that does symbolic summation using Gosper’s algorithm [3]. Here a call is made to the function Gosper of the Mathematica implementation of Gosper’s and Zeilberger’s algorithms written by Peter Paule and Markus Schorn. The current version 1.1 or updates can be received via e-mail request to peter.paule@risc.uni-linz.ac.at. This implemenation provides the user with the objects

    Zb, Gosper, RunMode, FileName, SolAmount, Fnk, GoRat, GoSol, Cert, DegBound, System, SystemDimension.

Also within the package HYP, all these objects work as described in the documentation of this implementation. Therefore the user is referred to this documentation and the description [5] in order to learn about the various features of these objects. The package HYP provides two additional objects, ZB and GOSPER. The rule GOSPER allows to apply Gosper’s algorithm directly to an expression containing a Sum or a hypergeometric series.

\underline{Usage}: Expr/.GOSPER[]
   or: Expr/.GOSPER[order].

(Runs Gosper’s algorithm trying an additional polynomial Ansatz of degree ’order’.)

\underline{Example(s)}:

In[1]:= Sum[1/(k*(k+1)),{k,1,n}]

66 Alphabetic List: Gzerl

          n
         \  1
Out[1]= ⟩ ---------
         /   k (1 + k)
         k=1
In[2]:= %/.GOSPER[]
    Peter Paule and Markus Schorn's implementation of the
                Gosper algorithm.  (Version 1.1)
                 1                          1
Out[2]= {SUM[---------, {k, 1, n}] == 1 - -----}
             k (1 + k)                    1 + n
In[3]:= F[{1,1},{3},1]/2
             1, 1
         F       ; 1
        2 1   3
Out[3]= --------------
              2
In[4]:= %/.GOSPER[]
    Peter Paule and Markus Schorn's implementation of the
                Gosper algorithm.  (Version 1.1)
                kk!
Out[4]= {SUM[---------, {kk, 0, ∞}] == 1}
             (2 + kk)!

\underline{See also}: ZB.

Gzerl \underline{Description}: Rule that splits GAMMA[List1,List2] into a quotient of products of Gamma functions.

\underline{Usage}: Expr/.Gzerl.

\underline{Example(s)}:

In[1]:= GAMMA[{a,b},{c,d}]

           a, b
Out[1]= Γ
           c, d

In[2]:= %/.Gzerl
        Γ(a) Γ(b)
Out[2]= ---------
        Γ(c) Γ(d)

\underline{See also}: paufl, pzerl, pzus, GAMMA, Gzus, Ers, PosListe.

Gzus

\underline{Description}: Rule that collects several Gammma functions GAMMA[xi] to an expression GAMMA[List1,List2].

\underline{Usage}: Expr/.Gzus.

\underline{Example(s)}:

In[1]:= GAMMA[a]*GAMMA[b]/GAMMA[c]/GAMMA[d]
        Γ(a) Γ(b)
Out[1]= ---------
        Γ(c) Γ(d)
In[2]:= %/.Gzus

           a, b
Out[2]= Γ
           c, d

\underline{See also}: paufl, pzerl, pzus, GAMMA, Gzerl, Ers, PosListe.

Hoch

\underline{Description}: Function that takes Gleichung to the Expr-th power.

\underline{Usage}: Hoch[Expr].

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n

                           (-a + c)
             a, -n                 n
Out[1]=  F        ; 1   == ---------
        2 1    c             (c)
                                n
In[2]:= Hoch[3]
                                     3
                            (-a + c)
             a, -n     3            n
Out[2]=  F        ; 1    == ----------
        2 1    c                  3
                              (c)
                                 n
In[3]:= Gleichung
                                     3
                            (-a + c)
             a, -n     3            n
Out[3]=  F        ; 1    == ----------
        2 1    c                  3
                              (c)
                                 n

\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Mal, Add, Div, Sub, GlTausche, Ers.

hypAttributes

\underline{Description}: Shows the current setup of the session. The setup can be changed by the switches P, FCancel, AmSTeX, AmSLaTeX, LaTeX, TeX, and TeXFV. The default-setup is shown in the following Example.

\underline{Usage}: hypAttributes.

\underline{Example(s)}:

In[1]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.

\underline{See also}: P, FCancel, AmSTeX, AmSLaTeX, LaTeX, TeX, TeXFV.

inv

\underline{Description}: Γ(a) → \frac{π}{sin(πa)} \frac{1}{Γ(1 − a)}.

\underline{Usage}: Expr/.inv.

\underline{Example(s)}:

In[1]:= GAMMA[2*a+1]
Out[1]= Γ(1 + 2 a)
In[2]:= %/.inv

        π Csc[(1 + 2 a) π]

Out[2]= ------------------
             Γ(-2 a)
\underline{See also}: inv, Ers, PosListe, ManipulationsListe.

LaTeX

\underline{Description}: Switch that changes the output of TeXForm to be usable with Plain-TEX and LaTEX. By default the output of TeXForm is usable with AMS-TEX.

\underline{Usage}: LaTeX.

\underline{Example(s)}:

In[1]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
In[2]:= TeXForm[F[{a,b},{c},z]]
Out[2]//TeXForm=
  {} {2} F {1} \!\left [ \matrix { a, b}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]
In[3]:= LaTeX
In[4]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
In[5]:= TeXForm[F[{a,b},{c},z]]

Out[5]//TeXForm=
  {} {2} F {1} \!\left [ \matrix { a, b\cr c} ; {\displaystyle z} \right ]
\underline{See also}: AmSTeX, AmSLaTeX, TeX, TeXMat, TeXFV.

Limes

\underline{Description}: Function for doing formal limits of hypergeometric expressions. If required for taking the limit, you will be asked whether or not the absolute value of some variable or expression is smaller than 1. You will be offered three options, [y|n|u]. If the absolute value of the variable is smaller than 1 then enter y, if it is greater than 1 then enter n, if you do not want to make an explicit declaration then enter u (for “undetermined”). Your decision, if explicit, is stored for the rest of your MATHEMATICA session. If you want to change your decision later, use

  AbsGreater, AbsSmaller, or AbsUndetermined, respectively.

\underline{Warning}: This function uses primitive algebraic techniques to do the limit. There is no check if taking the limit is actually allowed. So it is left to you to check the validity of a result of Limes.

\underline{Usage}: Limes[Expr, x->x0].

\underline{Example(s)}:

The derivation of the Vandermonde summation (see S2101) from the Pfaff–Saalschütz summation.

In[1]:= Sgl3201
Do you want to set values for the equation?  [y|n]:  n
                                          (-a + c)  (-b + c)
                   a, b, -n                       n         n
Out[1]=  F                       ; 1   == -------------------
        3 2  c, 1 + a + b - c - n         (c)  (-a - b + c)
                                             n             n
In[2]:= Limes[%,b-⟩Infinity]
                           (-a + c)
             a, -n                 n
Out[2]=  F        ; 1   == ---------
        2 1    c             (c)
                                n

The derivation of the Gauss summation (see S2103) from the Pfaff–Saalschütz summation.

In[3]:= Limes[%%,n-⟩Infinity]

             a, b  Γ(c) Γ(-a - b + c)
Out[3]=  F       ; 1   == -------------------
        2 1   c  Γ(-a + c) Γ(-b + c)

\underline{See also}: AbsGreater, AbsSmaller, AbsUndetermined, MinusOne.

lina1

\underline{Description}: \begin{matrix}(a)_{n} → a (a + 1)_{n−1}, \\ Γ(a) → (a − 1)Γ(a − 1). \end{matrix}

\underline{Usage}: Expr/.lina1.

\underline{Example(s)}:

In[1]:= p[a-1,m]
Out[1]= (-1 + a)
                m
In[2]:= %/.lina1
Out[2]= (-1 + a) (a)
                    -1 + m
In[3]:= 1/GAMMA[2*a]
          1
Out[3]= ------
        Γ(2 a)
In[4]:= %/.lina1
                  1
Out[4]= ----------------------
        (-1 + 2 a) Γ(-1 + 2 a)
\underline{See also}: lina2, linz, Ers, PosListe, ManipulationsListe.

lina2

\underline{Description}: (a)_{n} → (a + n − 1)(a)_{n−1}.

\underline{Usage}: Expr/.lina2.

\underline{Example(s)}:

In[1]:= p[a,m]
Out[1]= (a)
           m
In[2]:= %/.lina2
Out[2]= (-1 + a + m) (a)
                        -1 + m
\underline{See also}: lina1, linz, Ers, PosListe, ManipulationsListe.

linz

\underline{Description}: Rule that absorbs linear terms.

\underline{Usage}: Expr/.linz.

\underline{Example(s)}:

In[1]:= a*p[a+1,m]/(a/2+m-1)/p[a/2,m-1]
              a (1 + a)
                       m
Out[1]= ----------------------
              a       a
        (-1 + - + m) (-)
              2       2 -1 + m
In[2]:= %/.linz
               (a)
                  1 + m
Out[2]= ----------------------
              a       a
        (-1 + - + m) (-)
              2       2 -1 + m
In[3]:= %/.linz
        (a)
           1 + m
Out[3]= --------
           a
          (-)
           2 m
In[4]:= 1/(1-b)/GAMMA[b-1]
                1
Out[4]= -----------------
        (1 - b) Γ(-1 + b)
In[5]:= %/.linz
           1
Out[5]= -(----)
          Γ(b)
\underline{See also}: lina1, lina2, Ers, PosListe, ManipulationsListe.

LS

\underline{Description}: LS is the left-hand side in Gleichung.

\underline{Usage}: LS.

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
                           (-a + c)
             a, -n                 n
Out[1]=  F        ; 1   == ---------
        2 1    c             (c)
                                n
In[2]:= LS

             a, -n
Out[2]=  F        ; 1
        2 1    c

In[3]:= Add[1]
                                   (-a + c)
                 a, -n                     n
Out[3]= 1 +  F        ; 1   == 1 + ---------
            2 1    c                 (c)
                                        n
In[4]:= LS

                 a, -n
Out[4]= 1 +  F        ; 1
            2 1    c

In[5]:= LS=p[a,m]
Out[5]= (a)
           m

                    (-a + c)
                            n
Out[6]= (a)  == 1 + ---------
           m          (c)
                         n
\underline{See also}:  Gleichung, SumListe$gl, TransListe$gl, RS, Mal, Add, Div, Sub, Hoch, GlTausche, Ers,
  Subst.

Mal

\underline{Description}: Function that multiplies Gleichung by Expr.

\underline{Usage}: Mal[Expr].

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
                           (-a + c)
             a, -n                 n
Out[1]=  F        ; 1   == ---------
        2 1    c             (c)
                                n
In[2]:= Mal[p[c,n]]

             a, -n
Out[2]=  F        ; 1   (c)  == (-a + c)
        2 1    c           n            n

In[3]:= Gleichung

             a, -n
Out[3]=  F        ; 1   (c)  == (-a + c)
        2 1    c           n            n

\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Add, Div, Sub, Hoch, GlTausche, Ers.

ManipulationsListe

\underline{Description}: Gives a list of all available rules for manipulating factorial symbols and Gamma functions.

\underline{Usage}: ManipulationsListe.

MinusOne

\underline{Description}: : Rule for getting rid of expressions of the form (−1)^{N} where N is an even or odd integer.

\underline{Usage}: Expr/.MinusOne.
In[1]:= p[a,2*n]
Out[1]= (a)
           2 n
In[2]:= %/.trans
            2 n
Out[2]= (-1)    (1 - a - 2 n)
                             2 n
In[3]:= %/.MinusOne
Is 2 n even, odd, or neither of both?
[e|o|n]:  e
Out[3]= (1 - a - 2 n)
                     2 n

\underline{See also}: SimplifyP, Expandq, SumExpand.

neg1

\underline{Description}: (a)_{n} → 1/(a + n)_{−n}.

\underline{Usage}: Expr/.neg1.

\underline{Example(s)}:

In[1]:= p[a,-n]
Out[1]= (a)
           -n
In[2]:= %/.neg1
           1
Out[2]= --------
        (a - n)
               n

\underline{See also}: neg2, Ers, PosListe, ManipulationsListe.

neg2

\underline{Description}: (a)_{n} → (−1)^{n}/(1 − a)_{−n}.

\underline{Usage}: Expr/.neg2.

\underline{Example(s)}:

In[1]:= p[a,-n]
Out[1]= (a)
           -n
In[2]:= %/.neg2
             n
         (-1)
Out[2]= --------
        (1 - a)
               n

\underline{See also}: neg1, Ers, PosListe, ManipulationsListe.

p

\underline{Description}: p[x,n] is the factorial symbol (Pochhammer symbol) (x)_{n}. p[List1,List2,n] is also provided as the usual abbreviation for the quotient of factorial symbols (see [2]).

\underline{Usage}: p[x,n]
   or: p[List1,List2,n].

\underline{Example(s)}:

In[1]:= p[a,n]
Out[1]= (a)
           n
In[2]:= p[{a,b},{c,d},2*m]
        (a, b)
              2 m
Out[2]= ---------
        (c, d)
              2 m

\underline{See also}: F, GAMMA, Binomialp, Factorialp, P, paufl, pzerl, pzus, FFormat.

P

\underline{Description}: Is a switch that activates automatic evaluating of factorial symbols (Pochhammer symbols) p and hypergeometric series F, or makes it inactive, respectively. By default automatic evaluating is inactive.

\underline{Usage}: P.

\underline{Example(s)}:

In[1]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.
In[2]:= p[a,5]
Out[2]= (a)
           5
In[3]:= F[{a,b},{c},z]

             a, b
Out[3]=  F       ; z
        2 1   c

In[4]:= F[{-n,b},{c},z]

             -n, b
Out[4]=  F        ; z
        2 1    c

In[5]:= F[{-3,b},{c},z]

             -3, b
Out[5]=  F        ; z
        2 1    c

In[6]:= P
In[7]:= hypAttributes

Automatic evaluation of p and F is active.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.
In[8]:= p[a,5]
Out[8]= a (1 + a) (2 + a) (3 + a) (4 + a)
In[9]:= F[{a,b},{c},z]
Is -a a nonnegative integer?
[y|n]:    n
Is -b a nonnegative integer?
[y|n]:    n
A hypergeometric series is converted into a sum.
Enter a variable for the summation index:  k

          ∞ k

             z  (a)  (b)
         \  k    k
Out[9]= ⟩ ------------
         /    (1)  (c)
                 k    k
         k=0
In[10]:= F[{-n,b},{c},z]
Is n a nonnegative integer?
[y|n]:    y
Is -b a nonnegative integer?
[y|n]:    n
A hypergeometric series is converted into a sum.
Enter a variable for the summation index:  j
           n   j
              z  (b)  (-n)
          \  j     j
Out[10]= ⟩ -------------
          /     (1)  (c)
                   j    j
          j=0
In[11]:= F[{-3,b},{c},z]
Is -b a nonnegative integer?
[y|n]:    n
A hypergeometric series is converted into a sum.
Enter a variable for the summation index:  s

                                  2                      3
             3 b z   3 b (1 + b) z    b (1 + b) (2 + b) z
Out[11]= 1 - ----- + -------------- - --------------------
               c       c (1 + c)       c (1 + c) (2 + c)
In[12]:= P
In[13]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised hypergeometric series.

\underline{See also}: F, p, hypAttributes.

paufl

\underline{Description}: Rule that writes p[x,n] as the defining product Product[x+i,\{i,0,n-1\}], if n is an integer.

\underline{Usage}: Expr/.paufl.

\underline{Example(s)}:

In[1]:= p[a,-2]/p[b,3]*p[c,1]
        (a)   (c)
           -2    1
Out[1]= ----------
           (b)
              3
In[2]:= %/.paufl
                         c
Out[2]= -----------------------------------
        (-2 + a) (-1 + a) b (1 + b) (2 + b)
In[3]:= F[{a,b,c},{d,e,f},z]

             a, b, c
Out[3]=  F          ; z
        3 3  d, e, f

In[4]:= %/.C01

                   1, 1 + a, 1 + b, 1 + c
            z  F                         ; z   (a)  (b)  (c)
              4 4  2, 1 + d, 1 + e, 1 + f         1    1    1
Out[4]= 1 + -------------------------------------------------
                             (d)  (e)  (f)
                                1    1    1
In[5]:= %/.paufl

                         1, 1 + a, 1 + b, 1 + c
            a b c z  F                         ; z
                    4 4  2, 1 + d, 1 + e, 1 + f
Out[5]= 1 + ----------------------------------------
                             d e f

\underline{See also}: pzerl, pzus, p, Gzerl, Gzus, Ers, PosListe.

PosListe

\underline{Description}: Function that provides a list of subexpressions of Expr together with the respective positions in Expr. This helps to use controlled application of rules or functions by means of Ers.

\underline{Usage}: PosListe[Expr].

\underline{Example(s)}:

In[1]:= p[a,n]/p[1,n]*Sum[p[b,k]/p[c,k+1]*a^{∧}k,{k,0,Infinity}]

                ∞ k

                   a  (b)
               \  k
        (a)  ( ⟩ --------)
           n   /   (c)
                      1 + k
               k=0
Out[1]= --------------------
                (1)
                   n
In[2]:= PosListe[%]

                                          ∞ k
                                             a  (b)
           1  \  k
Out[2]= {{----, {{1}}}, {(a) , {{2}}}, { ⟩ --------, {{3}}}}
          (1)               n            /   (c)
             n                                  1 + k
                                         k=0
In[3]:= PosListe[%%,2]
Out[3]= {{-1, {{1, 2}}}, {a, {{2, 1}}}, {n, {{2, 2}}}, {{k, 0, ∞}, {{3, 2}}},
                         k
                        a  (b)
                              k
⟩  {(1) , {{1, 1}}}, {--------, {{3, 1}}}}
         n              (c)
                           1 + k
In[4]:= PosListe[%%%,3]
                                               k
Out[4]= {{0, {{3, 2, 2}}}, {1, {{1, 1, 1}}}, {a , {{3, 1, 1}}},
⟩  {k, {{3, 2, 1}}}, {n, {{1, 1, 2}}}, {∞, {{3, 2, 3}}},
                              1
⟩  {(b) , {{3, 1, 2}}}, {--------, {{3, 1, 3}}}}
         k                 (c)
                              1 + k

\underline{See also}: Ers, Subst.

PSort

\underline{Description}: Rule that orders the parameters of hypergeometric series F[List1,List2,z], of “multiple” Pochhammer symbols p[List1,List2,n], and of “multiple” Gamma functions GAMMA[List1,List2] in a standard order. For instance, this function can be used for a quick test if two expressions agree. It is recommended to apply Gzus and

  pzus[n] first.
\underline{Usage}: Expr/.PSort.

\underline{Example(s)}:

In[1]:= p[{c,c-a,b},{a-b-c,b+1},n]*GAMMA[{b-a,c},{a+b+c,-b}]*
        F[{b-c,b+a,a},{b,a},z]

                                                    (c, -a + c, b)
             b - c, a + b           -a + b, c                     n
Out[1]=  F               ; z Γ  -------------------
        2 1       b               a + b + c, -b   (a - b - c, 1 + b)
                                                                    n
In[2]:= %/.PSort
                                                    (b, c, -a + c)
             a + b, b - c           -a + b, c                     n
Out[2]=  F               ; z Γ  -------------------
        2 1       b               -b, a + b + c   (1 + b, a - b - c)
                                                                    n
\underline{See also}: SimplifyP, SumExpand, FEinf, FOrdne, FPerm, FTausche, F, V, p, GAMMA.

pzerl

\underline{Description}: Rule that splits p[List1,List2,n] into a quotient of products of factorial symbols (Pochhammer symbols).

\underline{Usage}: Expr/.pzerl.

\underline{Example(s)}:

In[1]:= p[{a,b},{c,d},n]
        (a, b)
              n
Out[1]= -------
        (c, d)
              n
In[2]:= %/.pzerl
        (a)  (b)
           n    n
Out[2]= ---------
        (c)  (d)
           n    n
\underline{See also}: paufl, pzus, p, Gzerl, Gzus, Ers, PosListe.

pzus

\underline{Description}: Rule that collects several factorial symbols (Pochhammer symbols) p[xi,n] to an expression

  p[List1,List2,n].
\underline{Usage}: Expr/.pzus[n].

\underline{Example(s)}:

In[1]:= p[a,n]*p[b,m]/p[c,n]/p[d,m]

        (a)  (b)
           n    m
Out[1]= ---------
        (c)  (d)
           n    m
In[2]:= %/.pzus[n]
             (a)
                n
        (b)  ----
           m (c)
                n
Out[2]= ---------
          (d)
             m
In[3]:= %/.pzus[m]
        (a)  (b)
           n    m
Out[3]= ---- ----
        (c)  (d)
           n    m
\underline{See also}: paufl, pzerl, p, Gzus, Gzerl, Ers, PosListe.

RS

\underline{Description}: RS is the right-hand side in Gleichung.

\underline{Usage}: RS.

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
                           (-a + c)
             a, -n                 n
Out[1]=  F        ; 1   == ---------
        2 1    c             (c)
                                n
In[2]:= RS
        (-a + c)
                n
Out[2]= ---------
          (c)
             n

In[3]:= Add[1]
                                   (-a + c)
                 a, -n                     n
Out[3]= 1 +  F        ; 1   == 1 + ---------
            2 1    c                 (c)
                                        n
In[4]:= RS
            (-a + c)
                    n
Out[4]= 1 + ---------
              (c)
                 n
In[5]:= RS=1/p[1,m]
         1
Out[5]= ----
        (1)
           m
In[6]:= Gleichung

                 a, -n          1
Out[6]= 1 +  F        ; 1   == ----
            2 1    c           (1)
                                  m
\underline{See also}:  Gleichung, SumListe$gl, TransListe$gl, LS, Mal, Add, Div, Sub, Hoch, GlTausche, Ers,
  Subst.

S1001

\underline{Description}: Summation formula ([7], Appendix (III.1)) in form of a rule. A q-analogue is HYPQ’s S1001.

1F_{0}[ \begin{matrix}a \\ − \end{matrix}; z] ⟶ (1 − z)^{−a}

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2101

\underline{Description}: Summation formula ([7], (1.7.7); Appendix (III.4)) in form of a rule. A q-analogue is HYPQ’s S2101.

2F_{1}[ \begin{matrix}a, −n \\ c \end{matrix}; 1] ⟶ \frac{(c − a)_{n}}{(c)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

Alphabetic List: S2106 85

S2103 \underline{Description}: Summation formula ([7], (1.7.6); Appendix (III.3)) in form of a rule. A q-analogue is HYPQ’s S2103.

2F_{1}[ \begin{matrix}a, b \\ c \end{matrix}; 1] ⟶ Γ [ \begin{matrix}c, c − a − b \\ c − a, −b + c \end{matrix} ]

\underline{Example(s)}:

In[1]:= F[{a,b},{2+a+b},1]
               a, b
Out[1]=  F            ; 1
        2 1  2 + a + b
In[2]:= %/.S2103
           2 + a + b, 2
Out[2]= Γ
           2 + b, 2 + a
\underline{See also}: S3201, SListe, SumListe, Ers, PosListe.

S2104 \underline{Description}: Summation formula ([7], (1.7.1.6), corrected, (2.3.2.9); Appendix (III.5)) in form of a rule. A q-analogue

is HYPQ’s S2104.

2F_{1}[ \begin{matrix}a, b \\ 1 + a − b \end{matrix}; −1] ⟶ Γ [ \begin{matrix}1 + \frac{a}{2}, 1 + a − b \\ 1 + a, 1 + \frac{a}{2} − b \end{matrix} ]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2105 \underline{Description}: Summation formula ([2], Ex. 1.6(i), q → 1) in form of a rule. A q-analogue is HYPQ’s S2105.

2F_{1}[ \begin{matrix}−\frac{n}{2}, \frac{1}{2} − \frac{n}{2} \\ \frac{1}{2} + b \end{matrix} ; 1] ⟶ \frac{2^{n}(b)_{n}}{(2b)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2106 \underline{Description}: Summation formula ([7], (1.5.21)) in form of a rule. q-Analogues are HYPQ’s S2106 and S3203.

2V_{1}(a; −; z) ⟶ (1 − z)^{−1−a} (1 + z)

S2131

\underline{Description}: Summation formula ([7], (1.7.1.8); Appendix (III.7)) in form of a rule. A q-analogue is HYPQ’s S2201.

2F_{1}[ \begin{matrix}a, 1 − a \\ b \end{matrix}; \frac{1}{2}] ⟶ Γ [ \begin{matrix}\frac{b}{2}, \frac{1}{2} + \frac{b}{2} \\ \frac{a}{2} + \frac{b}{2}, \frac{1}{2} − \frac{a}{2} + \frac{b}{2} \end{matrix} ]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S2132

\underline{Description}: Summation formula ([7], (1.7.1.9); Appendix (III.6)) in form of a rule. A q-analogue is HYPQ’s S2202.

2F_{1}[ \begin{matrix}2a, 2b \\ \frac{1}{2} + a + b \end{matrix}; \frac{1}{2}] ⟶ Γ [ \begin{matrix}\frac{1}{2}, \frac{1}{2} + a + b \\ \frac{1}{2} + a, \frac{1}{2} + b \end{matrix} ]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S3201

\underline{Description}: Summation formula ([7], (2.3.1.3); Appendix (III.2)) in form of a rule. A q-analogue is HYPQ’s S3201.

3F_{2}[ \begin{matrix}a, b, −n \\ c, 1 + a + b − c − n \end{matrix}; 1] ⟶ \frac{(c − a)_{n}(c − b)_{n}}{(c)_{n}(c − a − b)_{n}}

where n is a nonnegative integer.

\underline{Example(s)}:

In[1]:= F[{a,b,-n},{a+b-n,1},1]

               a, b, -n
Out[1]=  F               ; 1
        3 2  a + b - n, 1

In[2]:= %/.S3201
Is n a nonnegative integer?
[y|n]:    y
        (a - n)  (b - n)
               n        n
Out[2]= ------------------
        (a + b - n)  (-n)
                   n     n
In[3]:= %%/.S3201
Is n a nonnegative integer?
[y|n]:    n
Is -b a nonnegative integer?
[y|n]:    n
Is -a a nonnegative integer?
[y|n]:    n

               a, b, -n
Out[3]=  F               ; 1
        3 2  a + b - n, 1

In[4]:= F[{a,-m,-n},{a-m-n,1},1]

              a, -m, -n
Out[4]=  F               ; 1
        3 2  a - m - n, 1

In[5]:= %/.S3201
Is n a nonnegative integer?
[y|n]:    n
Is m a nonnegative integer?
[y|n]:    y
        (a - m)  (-m - n)
               m         m
Out[5]= ------------------
        (-m)  (a - m - n)
            m            m
\underline{See also}: S2103, SListe, SumListe, Ers, PosListe.

S3202

\underline{Description}: Summation formula ([7], (2.3.3.5); Appendix (III.8); terminated in the first variable) in form of a rule. A q-analogue is HYPQ’s S3202.

3F_{2}[ \begin{matrix}−2n, b, c \\ 1 − b − 2n, 1 − c − 2n \end{matrix}; 1] ⟶ \frac{(1)_{2n}(b)_{n}(c)_{n}(b + c)_{2n}}{(1)_{n}(b)_{2n}(c)_{2n}(b + c)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S3204

\underline{Description}: Summation formula ([2], Ex. 3.9, q → 1) in form of a rule. q-Analogues are HYPQ’s S3204 and S8704.

3F_{2}[ \begin{matrix}a, 1 + \frac{λ}{2}, b \\ \frac{λ}{2}, 1 − b + λ \end{matrix}; 1] ⟶ Γ [ \begin{matrix}λ, 1 − a + λ, −a − 2b + λ, 1 − b + λ \\ 1 + λ, −a + λ, −2b + λ, 1 − a − b + λ \end{matrix} ]

S3231 \underline{Description}: Summation formula ([7], (2.3.3.5); Appendix (III.8)) in form of a rule. A q-analogue is HYPQ’s S4301.

3F_{2}[ \begin{matrix}a, b, c \\ 1 + a − b, 1 + a − c \end{matrix}; 1] ⟶ Γ [ \begin{matrix}1 + a − b, 1 + a − c, 1 + \frac{a}{2}, 1 + \frac{a}{2} − b − c \\ 1 + a, 1 + \frac{a}{2} − b, 1 + \frac{a}{2} − c, 1 + a − b − c \end{matrix} ]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S3232 \underline{Description}: Summation formula ([7], (2.3.3.6); Appendix (III.9)) in form of a rule. A q-analogue is HYPQ’s S4302.

3F_{2}[ \begin{matrix}a, b, −n \\ 1 + a − b, 1 + a + n \end{matrix}; 1] ⟶ \frac{(1 + a)_{n}(1 + \frac{a}{2} − b)_{n}}{(1 + \frac{a}{2})_{n}(1 + a − b)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S3233 \underline{Description}: Summation formula ([7], (2.3.3.13); Appendix (III.23)) in form of a rule. A q-analogue is HYPQ’s S8702,

a terminating q-analogue is HYPQ’s S4303.

3F_{2}[ \begin{matrix}a, b, c \\ \frac{1+a+b}{2}, 2c \end{matrix}; 1] ⟶ Γ [ \begin{matrix}\frac{1}{2}, \frac{1}{2} + c, \frac{1}{2} + \frac{a}{2} + \frac{b}{2}, \frac{1}{2} − \frac{a}{2} − \frac{b}{2} + c \\ \frac{1}{2} + \frac{a}{2}, \frac{1}{2} + \frac{b}{2}, \frac{1}{2} − \frac{a}{2} + c, \frac{1}{2} − \frac{b}{2} + c \end{matrix} ]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S3234 \underline{Description}: Summation formula ([7], (2.3.3.14); Appendix (III.24)) in form of a rule. A q-analogue is HYPQ’s S8703,

a terminating q-analogue is HYPQ’s S4304.

3F_{2}[ \begin{matrix}a, 1 − a, c \\ d, 1 + 2c − d \end{matrix}; 1] ⟶ 2^{1−2c}πΓ [ \begin{matrix}d, 1 + 2c − d \\ \frac{1}{2} + \frac{a}{2} + c − \frac{d}{2}, \frac{a}{2} + \frac{d}{2}, 1 − \frac{a}{2} + c − \frac{d}{2}, \frac{1}{2} − \frac{a}{2} + \frac{d}{2} \end{matrix} ]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S3235 \underline{Description}: Summation formula ([7], (2.4.2.5); Appendix (III.16)) in form of a rule. A q-analogue is HYPQ’s S4308.

3F_{2}[ \begin{matrix}a, b, −n \\ 1 + a − b, 1 + 2b − n \end{matrix}; 1] ⟶ \frac{(1 + \frac{a}{2} − b)_{n}(a − 2b)_{n}(−b)_{n}}{(\frac{a}{2} − b)_{n}(1 + a − b)_{n}(−2b)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

3F_{2}[ \begin{matrix}a, b, c \\ d, 1 + a + b + c − d \end{matrix}; 1] ⟶ Γ [ \begin{matrix}1 + a − d, 1 + b − d, 1 + c − d, 1 + a + b + c − d \\ 1 − d, 1 + b + c − d, 1 + a + c − d, 1 + a + b − d \end{matrix} ]

− Γ [ \begin{matrix}d − 1, 1 + a − d, 1 + b − d, 1 + c − d, 1 + a + b + c − d \\ 1 − d, a, b, c, 2 + a + b + c − 2d \end{matrix} ]_{3}F_{2}[ \begin{matrix}1 + a − d, 1 + b − d, 1 + c − d \\ 2 − d, 2 + a + b + c − 2d \end{matrix}; 1]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S3291 \underline{Description}: Summation formula ([2], Ex. 2.9, q → 1) in form of a rule. A q-analogue is HYPQ’s S4361.

3F_{2}[ \begin{matrix}a, b, 1 + a − 2c \\ 1 + a − c, 2b \end{matrix}; 1] ⟶ Γ [ \begin{matrix}\frac{1}{2}, \frac{1}{2} + \frac{a}{2} − b, 1 + a − c, 1 + \frac{a}{2} − b − c \\ \frac{1}{2} + \frac{a}{2}, \frac{1}{2} − b, 1 + \frac{a}{2} − c, 1 + a − b − c \end{matrix} ]

−_{3}F_{2}[ \begin{matrix}1 − b, 1 + a − 2b, 2 + a − 2b − 2c \\ 2 + a − 2b − c, 2 − 2b \end{matrix}; 1] Γ [ \begin{matrix}\frac{1}{2} + \frac{a}{2} − b, 1 + \frac{a}{2} − b, −\frac{1}{2} + b, 1 + a − c, 1 + \frac{a}{2} − b − c, \frac{3}{2} + \frac{a}{2} − b − c \\ \frac{1}{2} + \frac{a}{2}, \frac{a}{2}, \frac{1}{2} − b, \frac{1}{2} + \frac{a}{2} − c, 1 + \frac{a}{2} − c, 2 + a − 2b − c \end{matrix} ]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4306 \underline{Description}: Summation formula ([7], (2.4.2.6); Appendix (III.17)) in form of a rule. A q-analogue is HYPQ’s S4306.

4F_{3}[ \begin{matrix}a, 1 + \frac{a}{2}, b, −n \\ \frac{a}{2}, 1 + a − b, 1 + 2b − n \end{matrix}; 1] ⟶ \frac{(a − 2b, −b)_{n}}{(1 + a − b, −2b)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4307 \underline{Description}: Summation formula ([7], (2.3.4.6); Appendix (III.10)) in form of a rule. A q-analogue is HYPQ’s S4307.

4V_{3}(a; b, c; −1) ⟶ Γ [ \begin{matrix}1 + a − b, 1 + a − c \\ 1 + a, 1 + a − b − c \end{matrix} ]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S4331 \underline{Description}: Summation formula ([7], Appendix (III.22)) in form of a rule. q-Analogues are HYPQ’s S5401 and S5501.

4V_{3}(a; b, c; 1) ⟶ Γ [ \begin{matrix}1 + a − b, 1 + a − c, \frac{1}{2} + \frac{a}{2}, \frac{1}{2} + \frac{a}{2} − b − c \\ 1 + a, 1 + a − b − c, \frac{1}{2} + \frac{a}{2} − b, \frac{1}{2} + \frac{a}{2} − c \end{matrix} ]

4F_{3}[ \begin{matrix}a, 1 + \frac{a}{2}, b, −n \\ \frac{a}{2}, 1 + a − b, 2 + 2b − n \end{matrix}; 1] ⟶ \frac{(\frac{1}{2} + \frac{a}{2} − b)_{n}(−1 + a − 2b)_{n}(−1 − b)_{n}}{(−\frac{1}{2} + \frac{a}{2} − b)_{n}(1 + a − b)_{n}(−1 − 2b)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S5431 \underline{Description}: Summation formula ([7], (2.3.4.5); Appendix (III.12)) in form of a rule. A q-analogue is HYPQ’s S6501.

5V_{4}(a; b, c, d; 1) ⟶ Γ [ \begin{matrix}1 + a − b, 1 + a − c, 1 + a − d, 1 + a − b − c − d \\ 1 + a, 1 + a − b − c, 1 + a − b − d, 1 + a − c − d \end{matrix} ]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S5432 \underline{Description}: Summation formula ([7], (2.3.4.6); Appendix (III.13)) in form of a rule. A q-analogue is HYPQ’s S6502.

5V_{4}(a; b, c, −n; 1) ⟶ \frac{(1 + a)_{n}(1 + a − b − c)_{n}}{(1 + a − b)_{n}(1 + a − c)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S6531 \underline{Description}: Summation formula ([8], (1.8)) in form of a rule. A q-analogue is HYPQ’s S8702.

6V_{5}(−\frac{1}{2} + \frac{a}{2} + \frac{b}{2} + c; a, b, c, \frac{1}{2} + \frac{a}{2} + \frac{b}{2} − c; −1) ⟶ Γ [ \begin{matrix}\frac{1}{2}, \frac{1}{2} + \frac{a}{2} + \frac{b}{2}, \frac{1}{2} + c, \frac{1}{2} + \frac{a}{2} − \frac{b}{2} + c, \frac{1}{2} − \frac{a}{2} + \frac{b}{2} + c \\ \frac{1}{2} + \frac{a}{2}, \frac{1}{2} + \frac{b}{2}, \frac{1}{2} − \frac{a}{2} + c, \frac{1}{2} − \frac{b}{2} + c, \frac{1}{2} + \frac{a}{2} + \frac{b}{2} + c \end{matrix} ]

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S6532 \underline{Description}: Summation formula ([7], Appendix (III.27), corrected; [8], (1.7)) in form of a rule. A q-analogue is

HYPQ’s S8703.

6V_{5}(a; b, 1 − b, d, 1 − d; −1) ⟶ 2^{2b}Γ [ \begin{matrix}1 + a − b, a + b, 1 + a − d, 1 + \frac{a}{2} + \frac{b}{2} − \frac{d}{2}, \frac{1}{2} + \frac{a}{2} + \frac{b}{2} + \frac{d}{2}, a + d \\ a, 1 + a, 1 + a + b − d, 1 + \frac{a}{2} − \frac{b}{2} − \frac{d}{2}, \frac{1}{2} + \frac{a}{2} − \frac{b}{2} + \frac{d}{2}, a + b + d \end{matrix} ]

7V_{6}(a; b, c, d, 1 + 2a − b − c − d + n, −n; 1) ⟶ \frac{(1 + a)_{n}(1 + a − b − c)_{n}(1 + a − b − d)_{n}(1 + a − c − d)_{n}}{(1 + a − b)_{n}(1 + a − c)_{n}(1 + a − d)_{n}(1 + a − b − c − d)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S7632 \underline{Description}: Summation formula ([7], (2.4.1.5); Appendix (III.19)) in form of a rule. A q-analogue is HYPQ’s S10901.

7V_{6}(a; b, \frac{1}{2} + b, a − 2b, 1 + 2a − 2b + n, −n; 1) ⟶ \frac{(1 + a, 1 + 2a − 4b)_{n}}{(1 + a − 2b, 1 + 2a − 2b)_{n}}

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

S7691 \underline{Description}: Summation formula ([7], (4.2.3.8); Appendix (III.32)) in form of a rule. A q-analogue is HYPQ’s S8761.

7V_{6}(a; b, c, d, e, 1 + 2a − b − c − d − e; 1)

⟶ Γ [ \begin{matrix}1 + a − c, 1 + a − d, 1 + a − e, −a + b + c + d + e, −a + b + c, −a + b + d, −a + b + e, 1 + a − c − d − e \\ 1 + a, −a + b, 1 + a − c − d, 1 + a − c − e, −a + b + d + e, 1 + a − d − e, −a + b + c + e, −a + b + c + d \end{matrix} ]

− Γ [ \begin{matrix}a − b, 1 + a − c, 1 + a − d, 1 + a − e, −a + b + c + d + e, −a + b + c, −a + b + d, −a + b + e, \\ 1 + a, c, d, e, 1 + 2a − b − c − d − e, −a + b, 1 + b − c, 1 + b − d, \end{matrix}

\begin{matrix}1 + a − c − d − e, 1 − a + 2b \\ 1 + b − e, −2a + 2b + c + d + e \end{matrix} ]_{7}V_{6}(−a + 2b; b, −a + b + c, −a + b + d, −a + b + e, 1 + a − c − d − e; 1)

\underline{See also}: S2103, S3201, SListe, SumListe, Ers, PosListe.

SchreibeZahl \underline{Description}: Variable that counts the number of expressions already written by using TeXMat. Can be reset by defining

a new value.

\underline{Usage}: SchreibeZahl=n Integer.

\underline{Example(s)}:

In[1]:= SchreibeZahl
Out[1]= 0
In[2]:= TeXMat[p[a,n],filename]
In[3]:= !type filename.m
A[1]:=
p[a, n]

In[3]:= !type filename.tex
A[1]:=
({ \textstyle a}) {n}
In[3]:= SchreibeZahl=4
Out[3]= 4
In[4]:= TeXMat[p[a,n],filename]
In[5]:= !type filename.m
A[1]:=
p[a, n]
A[5]:=
p[a, n]
In[5]:= !type filename.tex
A[1]:=
({ \textstyle a}) {n}
A[5]:=
({ \textstyle a}) {n}

\underline{See also}: TeXMat.

Sgl1001

\underline{Description}: Summation formula ([7], Appendix (III.1)) in form of an equation. It is the same summation as that in

  S1001.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2101

\underline{Description}: Summation formula ([7], Appendix (III.4)) in form of an equation. It is the same summation as that in

  S2101.

\underline{Example(s)}:

In[1]:= {a,c}
Out[1]= {a, c}
In[2]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
                           (-a + c)
             a, -n                 n
Out[2]=  F        ; 1   == ---------
        2 1    c             (c)
                                n

In[3]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  y
a=2*a
c=-b
n=n
                             (-2 a - b)
             2 a, -n                   n
Out[3]=  F          ; 1   == -----------
        2 1    -b               (-b)
                                    n
In[4]:= {a,c}
Out[4]= {a, c}
In[5]:= a=2*b
Out[5]= 2 b
In[6]:= Sgl2101
Some variables have a value.  Should the variables
{a, c, n} be cleared?  Do you want to set
values for the equation (v)?  [y|n|yv|nv]:  nv
a=a
c=c
n=n
                             (-2 b + c)
             2 b, -n                   n
Out[6]=  F          ; 1   == -----------
        2 1     c               (c)
                                   n
In[7]:= {a,c}
Out[7]= {2 b, c}
In[8]:= Sgl2101
Some variables have a value.  Should the variables
{a, c, n} be cleared?  Do you want to set
values for the equation (v)?  [y|n|yv|nv]:  y
                           (-a + c)
             a, -n                 n
Out[8]=  F        ; 1   == ---------
        2 1    c             (c)
                                n

94 Alphabetic List: Sgl2106

In[9]:= {a,c}
Out[9]= {a, c}
In[10]:= c=3*e
Out[10]= 3 e
In[11]:= Sgl2101
Some variables have a value.  Should the variables
{a, c, n} be cleared?  Do you want to set
values for the equation (v)?  [y|n|yv|nv]:  yv
a=w+2
c=d
n=n
                                (-2 + d - w)
              2 + w, -n                     n
Out[11]=  F            ; 1   == -------------
         2 1      d                 (d)
                                       n
In[12]:= {a,c}
Out[12]= {a, c}
\underline{See also}: SumListe$gl, Gleichung.

Sgl2103 \underline{Description}: Summation formula ([7], Appendix (III.3)) in form of an equation. It is the same summation as that in

  S2103.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2104 \underline{Description}: Summation formula ([7], Appendix (III.5)) in form of an equation. It is the same summation as that in

  S2104.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2105 \underline{Description}: Summation formula ([2], Ex. 1.6(i), q → 1) in form of an equation. It is the same summation as that in

  S2105.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2106 \underline{Description}: Summation formula ([7], (1.5.21)) in form of an equation. It is the same summation as that in S2106.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Alphabetic List: Sgl3233 95

Sgl2131 \underline{Description}: Summation formula ([7], Appendix (III.7)) in form of an equation. It is the same summation as that in

  S2131.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl2132 \underline{Description}: Summation formula ([7], Appendix (III.6)) in form of an equation. It is the same summation as that in

  S2132.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3201 \underline{Description}: Summation formula ([7], Appendix (III.2)) in form of an equation. It is the same summation as that in

  S3201.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3202 \underline{Description}: Summation formula ([7], Appendix (III.8), terminated in the first variable) in form of an equation. It is

the same summation as that in S3202.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3204 \underline{Description}: Summation formula ([2], Ex. 3.9, q → 1) in form of an equation. It is the same summation as that in

  S3204.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3231 \underline{Description}: Summation formula ([7], Appendix (III.8)) in form of an equation. It is the same summation as that in

  S3231.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3232 \underline{Description}: Summation formula ([7], Appendix (III.9)) in form of an equation. It is the same summation as that in

  S3232.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3233 \underline{Description}: Summation formula ([7], Appendix (III.23)) in form of an equation. It is the same summation as that in

  S3233.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

96 Alphabetic List: Sgl4332

Sgl3234 \underline{Description}: Summation formula ([7], Appendix (III.24)) in form of an equation. It is the same summation as that in

  S3234.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3235 \underline{Description}: Summation formula ([7], (2.4.2.5); Appendix (III.16)) in form of an equation. It is the same summation

as that in S3235.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3261 \underline{Description}: Summation formula ([7], Appendix (III.31)) in form of an equation. It is the same summation as that in

  S3261.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl3291 \underline{Description}: Summation formula ([2], Ex. 2.9, q → 1) in form of an equation. It is the same summation as that in

  S3291.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4306 \underline{Description}: Summation formula ([7], (2.4.2.6); Appendix (III.17)) in form of an equation. It is the same summation

as that in S4306.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4307 \underline{Description}: Summation formula ([7], (2.3.4.6); Appendix (III.10)) in form of an equation. It is the same summation

as that in S4307.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4331 \underline{Description}: Summation formula ([7], Appendix (III.22)) in form of an equation. It is the same summation as that in

  S4331.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl4332 \underline{Description}: Summation formula ([7], (2.4.2.7); Appendix (III.18)) in form of an equation. It is the same summation

as that in S4332.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Alphabetic List: SimplifyP 97

Sgl5431 \underline{Description}: Summation formula ([7], Appendix (III.12)) in form of an equation. It is the same summation as that in

  S5431.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl5432 \underline{Description}: Summation formula ([7], Appendix (III.13)) in form of an equation. It is the same summation as that in

  S5432.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl6531 \underline{Description}: Summation formula ([8], (1.8)) in form of an equation. It is the same summation as that in S6531.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl6532 \underline{Description}: Summation formula ([7], Appendix (III.27), corrected; [8], (1.7)) in form of an equation. It is the same

summation as that in S6532.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl7631 \underline{Description}: Summation formula ([7], Appendix (III.14)) in form of an equation. It is the same summation as that in

  S7631.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl7632 \underline{Description}: Summation formula ([7], (2.4.1.5); Appendix (III.19)) in form of an equation. It is the same summation

as that in S7632.

\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

Sgl7691 \underline{Description}: Summation formula ([7], Appendix (III.32)) in form of an equation. It is the same summation as that in

  S7691.
\underline{See also}: Sgl2101, SumListe$gl, Gleichung.

SimplifyP \underline{Description}: Rule that simplifies arguments in p, GAMMA, F, Sum, and expands exponents in powers.

\underline{Usage}: Expr/.SimplifyP.

\underline{Example(s)}:

In[1]:= p[a-(d-c)/2,(k-l)*2]/GAMMA[(f-g)^{∧}2]*F[{-(d+c)/2,-n},{-d/2-c/2},
        (t-1)/(1-t)]

             -c - d
             ------, -n
               2         -1 + t        c - d
         F             ; ------   (a + -----)
        2 1    -c   d    1 - t           2   2 (k - l)
               -- - -
               2    2
Out[1]= ----------------------------------------------
                                  2
                         Γ((f - g) )
In[2]:= %/.SimplifyP
Is c/2 + d/2 a nonnegative integer?
[y|n]:    n

             -n            c   d
         F     ; -1   (a + - - -)
        1 0  -             2   2 2 k - 2 l
Out[2]= ----------------------------------
                   2            2
                Γ(f  - 2 f g + g )
In[3]:= Simplify[%1]
             -c - d
             ------, -n
               2                   c   d
         F             ; -1   (a + - - -)
        2 1    -c   d              2   2 2 (k - l)
               -- - -
               2    2
Out[3]= ------------------------------------------
                                2
                       Γ((f - g) )
\underline{See also}: Expandq, MinusOne, SumExpand, PSort.

SListe

\underline{Description}: Rule that gives for a hypergeometric series a list of applicable summation formulas. Each entry of this list has the format {S⟨number⟩}, where S⟨number⟩ is the name of the summation in form of a rule which can be applied subsequently. You should be aware that SListe automatically applies FOrdne before checking which summation could be applied.

\underline{Important Note}: If the value returned by SListe is the empty set this does not mean that no summation can be applied. You always must remember that the list of summations included in this package is a list of basic summations. There are numerous special cases of these summations which are not contained in this list as a separate summation. The examples below should illustrate these remarks.

\underline{Usage}: Expr/.SListe.

\underline{Example(s)}:

In[1]:= F[{a,b},{c},1]

             a, b
Out[1]=  F       ; 1
        2 1   c

In[2]:= %/.SListe
Is -b a nonnegative integer?
[y|n]:    n
Is -a a nonnegative integer?
[y|n]:    n
Be sure to apply "FOrdne" before using the following information!
Out[2]= {{S2103}}
In[3]:= F[{a,-n},{c},1]

             a, -n
Out[3]=  F        ; 1
        2 1    c

In[4]:= %/.SListe
Is n a nonnegative integer?
[y|n]:    y
Be sure to apply "FOrdne" before using the following information!
Out[4]= {{S2101}, {S2103}}

Now we consider two examples illustrating the note above. Though none of the implemented summations can be applied, both series can be summed, the first by a special case of Dougall’s sum, the second by a special case of the very well-poised _{5}F_{4} sum. These facts are also observed by using this package.

In[5]:= F[{-n,b,c,1-b-c-n/2,-n},{1-b-n,1-c-n,b+c-n/2,1},1]
                                      n
                -n, b, c, 1 - b - c - -, -n
                                      2
Out[5]=  F                                     ; 1
        5 4                                n
             1 - b - n, 1 - c - n, b + c - -, 1

In[6]:= %/.SListe
Out[6]= {}
In[7]:= Sgl7631
Do you want to set values for the equation?  [y|n]:  y
a=-n
b=b
c=c
d=-n/2
n=n
                                      n
                -n, b, c, 1 - b - c - -, -n
                                      2
Out[7]=  F                                     ; 1   ==
        5 4                                n
             1 - b - n, 1 - c - n, b + c - -, 1
                                           2
                                        n            n
     (1 - n)  (1 - b - c - n)  (1 - b - -)  (1 - c - -)
            n                n          2 n          2 n
⟩  ---------------------------------------------------
                                    n                n
     (1 - b - n)  (1 - c - n)  (1 - -)  (1 - b - c - -)
                n            n      2 n              2 n
In[8]:= F[{1+a/2,1,b,c},{a/2,1+a-b,1+a-c},1]
                     a
                 1 + -, 1, b, c
                     2
Out[8]=  F                          ; 1
        4 3  a
             -, 1 + a - b, 1 + a - c
             2
In[9]:= %/.SListe
Out[9]= {}
In[10]:= %%/.FEinf
Add the parameter:  a

                         a
                  a, 1 + -, 1, b, c
                         2
Out[10]=  F                             ; 1
         5 4     a
              a, -, 1 + a - b, 1 + a - c
                 2
In[11]:= %/.SListe
Is -c a nonnegative integer?
[y|n]:    n
Be sure to apply "FOrdne" before using the following information!
Out[11]= {{S5431}}

\underline{See also}: TListe, FPerm, FTausche, SumListe.

Sub

\underline{Description}: Function that subtracts Expr from Gleichung.

\underline{Usage}: Sub[Expr].

\underline{Example(s)}:

In[1]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  n
                           (-a + c)
             a, -n                 n
Out[1]=  F        ; 1   == ---------
        2 1    c             (c)
                                n
In[2]:= Sub[p[a,n]/p[c-a,n]]
                            (a)            (a)        (-a + c)
             a, -n             n              n               n
Out[2]=  F        ; 1   - --------- == -(---------) + ---------
        2 1    c          (-a + c)       (-a + c)       (c)
                                  n              n         n

                            (a)            (a)        (-a + c)
             a, -n             n              n               n
Out[3]=  F        ; 1   - --------- == -(---------) + ---------
        2 1    c          (-a + c)       (-a + c)       (c)
                                  n              n         n
\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, Mal, Add, Div, Hoch, GlTausche, Ers.

Subst

\underline{Description}: Function that substitutes RS instead of LS at position Position in Expr. The parameters LS and RS are optional. If they are omitted, the right-hand side “RS” of Gleichung is substituted instead of the left-hand side

  “LS” of Gleichung.
\underline{Usage}: Subst[Expr,Position,LS,RS].

\underline{Example(s)}:

In[1]:= Sum[p[a,k]/p[b,k]/k!,{k,0,Infinity}]

∞

              (a)
         \  k
Out[1]= ⟩ -------
         /   k!  (b)
                   k
         k=0
In[2]:= Subst[%,{1},p[a,k]/p[b,k],F[{b-a,-k},{b},1]]

                  -a + b, -k
          ∞ F             ; 1
             2 1      b

\

Out[2]= ⟩ --------------------
         /            k!
         k=0
In[3]:= Sgl2101
Do you want to set values for the equation?  [y|n]:  y
a=b-a
c=b
n=k
                                (a)
             -a + b, -k            k
Out[3]=  F             ; 1   == ----
        2 1      b              (b)
                                   k

In[4]:= GlTausche
        (a)
           k         -a + b, -k
Out[4]= ---- ==  F             ; 1
        (b)     2 1      b
           k
In[5]:= Gleichung
        (a)
           k         -a + b, -k
Out[5]= ---- ==  F             ; 1
        (b)     2 1      b
           k
In[6]:= Subst[%1,{1}]

                  -a + b, -k
          ∞ F             ; 1
             2 1      b

\

Out[6]= ⟩ --------------------
         /            k!
         k=0
\underline{See also}: Gleichung, SumListe$gl, TransListe$gl, LS, RS, GlTausche, Ers, PosListe.

SumErw1

\underline{Description}: Rule that extends a Sum[] at the top.

∑_{k=l}^{n} Expr → ∑_{k=l}^{n+m} Expr − ∑_{k=n+1}^{n+m} Expr.

The parameter m has to be entered on request.

\underline{Usage}: Expr/.SumErw1.

\underline{Example(s)}:

In[1]:= Sum[a[k],{k,0,N}]
          N

\

Out[1]= ⟩ a[k]
         /
         k=0

104 Alphabetic List: SumErw2

In[2]:= %/.SumErw1
top-extend by:  3
                                          3 + N

\

Out[2]= -a[1 + N] - a[2 + N] - a[3 + N] + ⟩ a[k]
                                           /
                                           k=0
In[3]:= %%/.SumErw1
top-extend by:  M
        M + N        M + N

\  \

Out[3]= ⟩ a[k] -  ⟩ a[k]
         /            /
         k=0        k=1 + N
\underline{See also}: SumErw2, SumZerl, SumShift, SumTausche, SumRegeln, SumUmkehr, Ers, PosListe.

SumErw2 \underline{Description}: Rule that extends a Sum[] at the bottom.

∑_{k=l}^{n} Expr → ∑_{k=l−m}^{n} Expr − ∑_{k=l−m}^{l−1} Expr.

The parameter m has to be entered on request.

\underline{Usage}: Expr/.SumErw2.

\underline{Example(s)}:

In[1]:= Sum[a[k],{k,0,N}]
          N

\

Out[1]= ⟩ a[k]
         /
         k=0
In[2]:= %/.SumErw2
bottom-extend by:  3

Alphabetic List: SumExpand 105

                                   N

\

Out[2]= -a[-3] - a[-2] - a[-1] + ⟩ a[k]
                                  /
                                 k=-3
In[3]:= %%/.SumErw2
bottom-extend by:  M
           -1            N

\  \

Out[3]= -( ⟩ a[k]) + ⟩ a[k]
           /            /
          k=-M         k=-M
\underline{See also}: SumErw1, SumZerl, SumShift, SumTausche, SumRegeln, SumUmkehr, Ers, PosListe.

SumExpand \underline{Description}: Rule that expands Sums.

\underline{Usage}: Expr/.SumExpand.

\underline{Example(s)}:

In[1]:= Sum[(x[k]-y[k])^{∧}2,{k,0,(m+n)/2}]
        m + n
        -----
          2
                          2
Out[1]= \ (x[k] - y[k])

⟩

         /
         k=0
In[2]:= %/.SumExpand

        m   n        m   n               m   n
        - + -        - + -               - + -
        2   2        2   2               2   2
                 2                                2
Out[2]= \ x[k]  + \ -2 x[k] y[k] + \ y[k]

⟩  ⟩  ⟩

         /            /                   /
         k=0          k=0                 k=0
\underline{See also}: SimplifyP, Expandq, MinusOne, PSort.

SumF

\underline{Description}: Rule that transforms a Sum[] into hypergeometric notation, if possible. If the upper bound is not Infinity you have to apply SumInfinity first (if allowed).

\underline{Usage}: Expr/.SumF.

\underline{Example(s)}:

In[1]:= Sum[p[-n,k]/p[1,k]*a^{∧}k,{k,0,Infinity}]

          ∞ k

             a  (-n)
         \  k
Out[1]= ⟩ --------
         /     (1)
                  k
         k=0
In[2]:= %/.SumF

             -n
Out[2]=  F     ; a
        1 0  -

In[3]:= Sum[(k+2)*p[{-n,a},{b,c,1},k+1]*z^{∧}k,{k,0,Infinity}]

∞

                         (-n, a)
         \  k         1 + k
Out[3]= ⟩ (2 + k) z  --------------
         /              (b, c, 1)
                                 1 + k
         k=0
In[4]:= %/.SumF

               3, 1 + a, 1 - n, 1
        2  F                     ; z   (a)  (-n)
          4 4  2, 2, 1 + b, 1 + c         1     1
Out[4]= -----------------------------------------
                     (1)  (b)  (c)
                        1    1    1
\underline{See also}: SumRegeln, SumInfinity, F, FSum, Ers, PosListe.

SumInfinity

\underline{Description}: Rule that changes the upper bound of a Sum[] to Infinity.

\underline{Usage}: Expr/.SumInfinity.

\underline{Example(s)}:

In[1]:= Sum[p[-n,k]/p[1,k]*a^{∧}k,{k,0,n}]
          n   k
             a  (-n)
         \  k
Out[1]= ⟩ --------
         /     (1)
                  k
         k=0
In[2]:= %/.SumInfinity

          ∞ k

             a  (-n)
         \  k
Out[2]= ⟩ --------
         /     (1)
                  k
         k=0
\underline{See also}: SumF, Ers, PosListe.

SumListe

\underline{Description}: List of all summation formulas.

\underline{Usage}: SumListe.
\underline{See also}: SumListe$gl, SListe.

108 Alphabetic List: SumSammle

SumListe$gl \underline{Description}: List of all summation formulas.

\underline{Usage}: SumListe$gl.
\underline{See also}: SumListe.

SumRegeln \underline{Description}: Rule that transforms the expressions in a Sum[] into a form that could also be expressed in hypergeometric

notation. This is useful, if you want to convert a Sum[] into hypergeometric notation but without using the F[]- notation. In particular, expressions of the form (−1)^{dk}, where d is an integer and k is the summation index, will simplify.

\underline{Usage}: Expr/.SumRegeln.

\underline{Example(s)}:

In[1]:= Sum[Binomial[n,i]*Binomial[m,k-i],{i,0,Infinity}]

∞

             (        ) (   )
         \ (   m    ) ( n )
Out[1]= ⟩ (        ) (   )
         /   ( -i + k ) ( i )
             (        ) (   )
         i=0
In[2]:= %/.SumRegeln

∞

                              (-k)  (-n)
                       \  i     i
        (1 - k + m)  ( ⟩ -----------------)
                   k   /   (1)  (1 - k + m)
                              i            i
                       i=0
Out[2]= -------------------------------------
                        (1)
                           k
\underline{See also}: F, SumF, FSum, MinusOne, Ers, PosListe.

SumSammle \underline{Description}: Rule that causes all terms of an expression Expr, which involves a Sum[] to be put into the Sum[].

\underline{Usage}: Expr/.SumSammle.

\underline{Example(s)}:

In[1]:= p[a,n]/p[b,m]*(-1)^{∧}n*Sum[1/p[1,k],{k,0,Infinity}]

∞

            n  \  1
        (-1)  (a)  ( ⟩ ----)

                 n   /   (1)
                            k
                     k=0
Out[1]= ----------------------
                 (b)
                    m
In[2]:= %/.SumSammle

          ∞  n

             (-1)  (a)
         \  n
Out[2]= ⟩ ----------
         /   (1)  (b)
                k    m
         k=0
\underline{See also}: SumRegeln, SumErw1, SumErw2, SumZerl, SumShift, SumTausche, pzus, Gzus, Ers, PosListe.

SumShift

\underline{Description}: Rule that shifts the index in a Sum[].

∑_{k=l}^{n} Expr(k) → ∑_{k=l−m}^{n−m} Expr(k + m).

The parameter m has to be entered on request.

\underline{Usage}: Expr/.SumShift.

\underline{Example(s)}:

In[1]:= Sum[a[k],{k,3,N}]
          N

\

Out[1]= ⟩ a[k]
         /
         k=3
In[2]:= %/.SumShift
shift summation index by:  3

        -3 + N

\

Out[2]= ⟩ a[3 + k]
         /
         k=0
\underline{See also}: SumErw1, SumErw2, SumShift, SumZerl, SumRegeln, SumUmkehr, Ers, PosListe.

SumTausche

\underline{Description}: Rule that exchanges summations. You should apply SumSammle before applying SumTausche.

∑_{k_{1}=l_{1}}^{n_{1}} ∑_{k_{2}=l_{2}}^{n_{2}} Expr → ∑_{k_{2}=l_{2}}^{n_{2}} ∑_{k_{1}=l_{1}}^{n_{1}} Expr.

\underline{Usage}: Expr/.SumTausche.

\underline{Example(s)}:

In[1]:= Sum[Sum[Binomial[n,k+l],{k,0,n1}],{l,0,n2}]
         n2   n1
                  (       )
         \  \ (   n   )
Out[1]= ⟩  ⟩ (       )
         /    /   ( k + l )
                  (       )
         l=0  k=0
In[2]:= %/.SumTausche
         n1   n2
                  (       )
         \  \ (   n   )
Out[2]= ⟩  ⟩ (       )
         /    /   ( k + l )
                  (       )
         k=0  l=0
\underline{See also}: SumErw1, SumErw2, SumSammle, SumShift, SumZerl, SumRegeln, SumUmkehr, Ers, PosListe.

SumUmkehr

\underline{Description}: Rule that reverses the order of summation. SumUmkehr applies to Sum[] as well as F[].

\underline{Usage}: Expr/.SumUmkehr.

\underline{Example(s)}:

In[1]:= F[{a,-n},{b},z]

             a, -n
Out[1]=  F        ; z
        2 1    b

In[2]:= %/.SumUmkehr
Is -a a nonnegative integer?
[y|n]:    n
Is n a nonnegative integer?
[y|n]:    y
                                         (a)
            n  n      -n, 1 - b - n  1      n
Out[2]= (-1)  z   F                ; -   ----
                 2 1    1 - a - n    z   (b)
                                            n
In[3]:= Sum[p[-n,k]/p[1,k],{k,0,Infinity}]

∞

             (-n)
         \  k
Out[3]= ⟩ -----
         /   (1)
                k
         k=0
In[4]:= %/.SumUmkehr
Is n a nonnegative integer?
[y|n]:    y
           n
              (-n)
          \  k
        ( ⟩ -----) (-n)
          /   (1)        n
                 k
          k=0
Out[4]= ------------------
               (1)
                  n

\underline{See also}: SumErw1, SumErw2, SumZerl, SumShift, SumTausche, SumRegeln, Ers, PosListe.

112 Alphabetic List: T2103

SumZerl \underline{Description}: Rule that splits a Sum[] into two parts.

∑_{k=l}^{n} Expr → ∑_{k=l}^{l+m−1} Expr + ∑_{k=l+m}^{n} Expr.

The parameter m has to be entered on request.

\underline{Usage}: Expr/.SumZerl.

\underline{Example(s)}:

In[1]:= Sum[a[k],{k,0,N}]
          N

\

Out[1]= ⟩ a[k]
         /
         k=0
In[2]:= %/.SumZerl
bottom-split by:  3
                               N

\

Out[2]= a[0] + a[1] + a[2] + ⟩ a[k]
                              /
                              k=3
In[3]:= %%/.SumZerl
bottom-split by:  M
        -1 + M         N

\  \

Out[3]= ⟩ a[k] + ⟩ a[k]
         /            /
         k=0          k=M
\underline{See also}: SumErw1, SumErw2, SumShift, SumTausche, SumRegeln, SumUmkehr, Ers, PosListe.

T2103 \underline{Description}: Transformation formula ([7], (1.3.15)) in form of a rule. A q-analogue is HYPQ’s T2103.

2F_{1}[ \begin{matrix}a, b \\ c \end{matrix}; z] ⟶ (1 − z)^{c−a−b}_{2}F_{1}[ \begin{matrix}c − a, c − b \\ c \end{matrix}; z]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2104 \underline{Description}: Transformation formula ([7], (1.7.1.3)) in form of a rule. A q-analogue is HYPQ’s T2104.

2F_{1}[ \begin{matrix}a, b \\ c \end{matrix}; z] ⟶ (1 − z)^{−a}_{2}F_{1}[ \begin{matrix}a, c − b \\ c \end{matrix}; −\frac{z}{1 − z} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2106 \underline{Description}: Transformation formula ([7], (1.7.1.3), sum reversed at the right-hand side) in form of a rule. A q-analogue

is HYPQ’s T2106.

2F_{1}[ \begin{matrix}a, −n \\ c \end{matrix}; z] ⟶ z^{n} \frac{(c − a)_{n}}{(c)_{n}}_{2}F_{1}[ \begin{matrix}−n, 1 − c − n \\ 1 + a − c − n \end{matrix}; −\frac{1 − z}{z} ]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2107 \underline{Description}: Transformation formula ([7], (1.8.10), terminating form) in form of a rule. A q-analogue is HYPQ’s

  T2107.

2F_{1}[ \begin{matrix}a, −n \\ c \end{matrix}; z] ⟶ \frac{(c − a)_{n}}{(c)_{n}}_{2}F_{1}[ \begin{matrix}−n, a \\ 1 + a − c − n \end{matrix}; 1 − z]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2110 \underline{Description}: Transformation formula ([6], (3.2)) in form of a rule. A q-analogue is HYPQ’s T2110.

2F_{1}[ \begin{matrix}a, b \\ 1 + a − b \end{matrix}; z] ⟶ (1 + z)^{−a}_{2}F_{1}[ \begin{matrix}\frac{a}{2}, \frac{1}{2} + \frac{a}{2} \\ 1 + a − b \end{matrix}; \frac{4z}{(1 + z)^{2}} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2112 \underline{Description}: Transformation formula ([6], (5.10)) in form of a rule. A q-analogue is HYPQ’s T2112.

2F_{1}[ \begin{matrix}a, \frac{1}{2} + a \\ \frac{1}{2} + b \end{matrix}; z^{2}] ⟶ (1 − z)^{−2a}_{2}F_{1}[ \begin{matrix}2a, b \\ 2b \end{matrix}; \frac{2z}{−1 + z} ]

114 Alphabetic List: T2136

T2131 \underline{Description}: Transformation formula ([7], (1.8.10), terminating form, sum reversed at the right-hand side) in form of

a rule. A q-analogue is HYPQ’s T3202.

2F_{1}[ \begin{matrix}a, −n \\ c \end{matrix}; z] ⟶ (1 − z)^{n} \frac{(a)_{n}}{(c)_{n}}_{2}F_{1}[ \begin{matrix}−n, c − a \\ 1 − a − n \end{matrix}; \frac{1}{1 − z} ]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2132 \underline{Description}: Transformation formula ([7], (2.5.7)) in form of a rule. A q-analogue is HYPQ’s T4307.

2F_{1}[ \begin{matrix}a, b \\ \frac{1}{2} + a + b \end{matrix}; z]^{2} ⟶ _{3}F_{2}[ \begin{matrix}2a, 2b, a + b \\ 2a + 2b, \frac{1}{2} + a + b \end{matrix}; z]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2133 \underline{Description}: Transformation formula ([6], (5.12)) in form of a rule. A q-analogue is HYPQ’s T2202.

2F_{1}[ \begin{matrix}a, b \\ \frac{1}{2} + a + b \end{matrix}; \frac{z^{2}}{−1 + z^{2}} ] ⟶ \frac{(1 − z)^{a}}{(1 + z)^{a}}_{2}F_{1}[ \begin{matrix}2a, a + b \\ 2a + 2b \end{matrix}; \frac{2z}{1 + z} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2134 \underline{Description}: Transformation formula ([1], Ex. 4.(iii), p. 97, reversed) in form of a rule. A q-analogue is HYPQ’s

  T3211.

2F_{1}[ \begin{matrix}a, b \\ \frac{1}{2} + \frac{a}{2} + \frac{b}{2} \end{matrix} ; z] ⟶ _{2}F_{1}[ \begin{matrix}\frac{a}{2}, \frac{b}{2} \\ \frac{1}{2} + \frac{a}{2} + \frac{b}{2} \end{matrix} ; 4z (1 − z)]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2135 \underline{Description}: Transformation formula ([1], Ex. 4.(iii), p. 97) in form of a rule. A q-analogue is HYPQ’s T3212.

2F_{1}[ \begin{matrix}a, b \\ \frac{1}{2} + a + b \end{matrix}; z] ⟶ _{2}F_{1}[ \begin{matrix}2a, 2b \\ \frac{1}{2} + a + b \end{matrix}; \frac{1 − \sqrt{1 − z}}{2} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2136 \underline{Description}: Transformation formula ([6], (5.10), reversed) in form of a rule. A q-analogue is HYPQ’s T3213.

2F_{1}[ \begin{matrix}a, b \\ 2b \end{matrix}; z] ⟶ (\frac{2}{2 − z} )^{a} _{2}F_{1}[ \begin{matrix}\frac{a}{2}, \frac{1}{2} + \frac{a}{2} \\ \frac{1}{2} + b \end{matrix} ; \frac{z^{2}}{(z − 2)^{2}} ]

Alphabetic List: T2163 115

T2137 \underline{Description}: Transformation formula ([6], (5.12), reversed) in form of a rule. A q-analogue is HYPQ’s T3214.

2F_{1}[ \begin{matrix}a, b \\ 2b \end{matrix}; z] ⟶ (1 − z)^{− \frac{a}{2}} _{2}F_{1}[ \begin{matrix}\frac{a}{2}, −\frac{a}{2} + b \\ \frac{1}{2} + b \end{matrix}; \frac{z^{2}}{4 (z − 1)}]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2138 \underline{Description}: Transformation formula ([6], (3.31), reversed) in form of a rule. A q-analogue is HYPQ’s T3215.

2F_{1}[ \begin{matrix}a, 1 − a \\ c \end{matrix}; z] ⟶ (1 − z)^{c−1}_{2}F_{1}[ \begin{matrix}\frac{c}{2} − \frac{a}{2}, \frac{a}{2} + \frac{c}{2} − \frac{1}{2} \\ c \end{matrix} ; 4z (1 − z)]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2139 \underline{Description}: Transformation formula ([6], (3.31)) in form of a rule. A q-analogue is HYPQ’s T3216.

2F_{1}[ \begin{matrix}c, −\frac{1}{2} + a − c \\ a \end{matrix}; z] ⟶ \frac{(1 + \sqrt{1 − z})}{2}^{1−a}_{2}F_{1}[ \begin{matrix}a − 2c, 1 − a + 2c \\ a \end{matrix}; \frac{1 − \sqrt{1 − z}}{2} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2140 \underline{Description}: Transformation formula ([6], (3.2), reversed) in form of a rule. A q-analogue is HYPQ’s T8707.

2F_{1}[ \begin{matrix}a, \frac{1}{2} + a \\ b \end{matrix}; \frac{4z}{(1 + z)^{2}} ] ⟶ (1 + z)^{2a}_{2}F_{1}[ \begin{matrix}2a, 1 + 2a − b \\ b \end{matrix}; z]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2141 \underline{Description}: Transformation formula ([2], (3.4.8), q → 1, reversed) in form of a rule. A q-analogue is HYPQ’s T8708.

2F_{1}[ \begin{matrix}a, \frac{1}{2} + a \\ b \end{matrix}; \frac{4z}{(1 + z)^{2}} ] ⟶ \frac{(1 + z)^{2a}}{1 − z}_{3}V_{2}(−1 + 2a; 2a − b; z)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2163 \underline{Description}: Transformation formula ([2], Ex 3.8, q → 1; [7], pp. 36/37) in form of a rule. A q-analogue is HYPQ’s

  T2163.

2F_{1}[ \begin{matrix}a, b \\ c \end{matrix}; z] ⟶ (1 − z)^{−b}Γ [ \begin{matrix}a − b, c \\ a, −b + c \end{matrix} ]_{2}F_{1}[ \begin{matrix}b, −a + c \\ 1 − a + b \end{matrix}; \frac{1}{1 − z} ] + (1 − z)^{−a}Γ [ \begin{matrix}−a + b, c \\ b, −a + c \end{matrix} ]_{2}F_{1}[ \begin{matrix}a, −b + c \\ 1 + a − b \end{matrix}; \frac{1}{1 − z} ]

T2191 \underline{Description}: Transformation formula ([7], (1.8.10)) in form of a rule.

2F_{1}[ \begin{matrix}a, b \\ c \end{matrix}; z] ⟶ (1 − z)^{c−a−b}Γ [ \begin{matrix}c, a + b − c \\ a, b \end{matrix} ]_{2}F_{1}[ \begin{matrix}c − b, c − a \\ 1 − a − b + c \end{matrix}; 1 − z]

+ Γ [ \begin{matrix}c, c − a − b \\ c − b, c − a \end{matrix} ]_{2}F_{1}[ \begin{matrix}a, b \\ 1 + a + b − c \end{matrix}; 1 − z]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T2192 \underline{Description}: Transformation formula ([2], Ex 3.8, q → 1, reversed; [7], pp. 36/37) in form of a rule. A q-analogue is

HYPQ’s T3269.

2F_{1}[ \begin{matrix}a, b \\ c \end{matrix}; z] ⟶ z^{−a}Γ [ \begin{matrix}1 + a − c, 1 + b − c \\ 1 − c, 1 + a + b − c \end{matrix} ]_{2}F_{1}[ \begin{matrix}a, 1 + a − c \\ 1 + a + b − c \end{matrix}; \frac{z − 1}{z}]

− z^{1−c}Γ [ \begin{matrix}1 + a − c, 1 + b − c, −1 + c \\ a, b, 1 − c \end{matrix} ]_{2}F_{1}[ \begin{matrix}1 + a − c, 1 + b − c \\ 2 − c \end{matrix}; z]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3204 \underline{Description}: Transformation formula ([1], Ex. 7, p. 98) in form of a rule. A q-analogue is HYPQ’s T3204.

3F_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; 1] ⟶ Γ [ \begin{matrix}e, −a − b − c + d + e \\ −a + e, −b − c + d + e \end{matrix} ]_{3}F_{2}[ \begin{matrix}a, −b + d, −c + d \\ d, −b − c + d + e \end{matrix}; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3205 \underline{Description}: Transformation formula ([7], (2.3.3.7)) in form of a rule. A q-analogue is HYPQ’s T3205.

3F_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; 1] ⟶ Γ [ \begin{matrix}d, e, −a − b − c + d + e \\ b, −a − b + d + e, −b − c + d + e \end{matrix} ]_{3}F_{2}[ \begin{matrix}−b + d, −b + e, −a − b − c + d + e \\ −a − b + d + e, −b − c + d + e \end{matrix}; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3206 \underline{Description}: Transformation formula ([1], Ex. 7, p. 98, terminating form) in form of a rule. A q-analogue is HYPQ’s

  T3206.

3F_{2}[ \begin{matrix}a, b, −n \\ d, e \end{matrix}; 1] ⟶ \frac{(−a − b + d + e)_{n}}{(e)_{n}}_{3}F_{2}[ \begin{matrix}−n, −a + d, −b + d \\ d, −a − b + d + e \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3207 \underline{Description}: Transformation formula (Thomae [1879] in [2] (3.1.1)) in form of a rule. A q-analogue is HYPQ’s T3207.

3F_{2}[ \begin{matrix}a, b, −n \\ d, e \end{matrix}; 1] ⟶ \frac{(−b + e)_{n}}{(e)_{n}}_{3}F_{2}[ \begin{matrix}−n, b, −a + d \\ d, 1 + b − e − n \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3217 \underline{Description}: Transformation formula ([1] 4.4(2), reversed) in form of a rule. A q-analogue is HYPQ’s T3217.

3F_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; 1] ⟶ Γ [ \begin{matrix}−a + d + e, −a − b − c + d + e \\ −a − b + d + e, −a − c + d + e \end{matrix} ]_{6}V_{5}(−1 − a + d + e; −a + e, −a + d, b, c; −1)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3231 \underline{Description}: Transformation formula ([2], Appendix (III.21), q ↑ 1) in form of a rule. A q-analogue is HYPQ’s T4305.

3F_{2}[ \begin{matrix}2a, 2b, c \\ \frac{1}{2} + a + b, d \end{matrix}; 1] ⟶ _{4}F_{3}[ \begin{matrix}a, b, c, −c + d \\ \frac{1}{2} + a + b, \frac{d}{2}, \frac{1+d}{2} \end{matrix} ; 1]

provided both series terminate.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3232 \underline{Description}: Transformation formula ([1], Ex. 4.(iv), p.97, reversed, first form) in form of a rule. A q-analogue is

HYPQ’s T5469. A terminating q-analogue is HYPQ’s T5404.

3F_{2}[ \begin{matrix}\frac{a}{2}, \frac{1+a}{2}, 1 + a − b − c \\ 1 + a − b, 1 + a − c \end{matrix}; \frac{−4 z}{(1 − z)^{2}} ] ⟶ (1 − z)^{a} _{3}F_{2}[ \begin{matrix}a, b, c \\ 1 + a − b, 1 + a − c \end{matrix}; z]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3233 \underline{Description}: Transformation formula ([1], Ex. 4.(iv), p.97, reversed, second form) in form of a rule. A q-analogue is

HYPQ’s T5469. A terminating q-analogue is HYPQ’s T5404.

3F_{2}[ \begin{matrix}\frac{a}{2}, \frac{1+a}{2}, 1 + a − b − c \\ 1 + a − b, 1 + a − c \end{matrix}; \frac{−4 z}{(1 − z)^{2}} ] ⟶ (1 − \frac{1}{z})^{a} _{3}F_{2}[ \begin{matrix}a, b, c \\ 1 + a − b, 1 + a − c \end{matrix}; \frac{1}{z}]

T3234 \underline{Description}: Transformation formula ([7], (2.5.7), reversed) in form of a rule. A q-analogue is HYPQ’s T5405.

3F_{2}[ \begin{matrix}2 a, 2 b, a + b \\ 2 a + 2 b, \frac{1}{2} + a + b \end{matrix}; z] ⟶ _{2}F_{1}[ \begin{matrix}a, b \\ \frac{1}{2} + a + b \end{matrix}; z]^{2}

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3235 \underline{Description}: Transformation formula ([2], Ex 3.4, q → 1, reversed) in form of a rule. A q-analogue is HYPQ’s T4201.

3F_{2}[ \begin{matrix}a, b, −n \\ d, 2b \end{matrix}; 2] ⟶ \frac{(−a + d)_{n}}{(d)_{n}}_{4}F_{3}[ \begin{matrix}\frac{a}{2}, \frac{1}{2} + \frac{a}{2}, \frac{1}{2} − \frac{n}{2}, \frac{−n}{2} \\ \frac{1}{2} + \frac{a}{2} − \frac{d}{2} − \frac{n}{2}, 1 + \frac{a}{2} − \frac{d}{2} − \frac{n}{2}, \frac{1}{2} + b \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3236 \underline{Description}: Transformation formula ([2], (3.4.8), q → 1) in form of a rule. A q-analogue is HYPQ’s T4311.

3V_{2}(a; b; z) ⟶ \frac{(1 − z)}{(1 + z)^{1+a}}_{2}F_{1}[ \begin{matrix}\frac{1}{2} + \frac{a}{2}, 1 + \frac{a}{2} \\ 1 + a − b \end{matrix} ; \frac{4z}{(1 + z)^{2}} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3237 \underline{Description}: Transformation formula ([2], (3.10.4), q → 1, reversed) in form of a rule. A q-analogue is HYPQ’s T5403.

3F_{2}[ \begin{matrix}x, y, −n \\ b, a \end{matrix}; 1] ⟶ \frac{(a − x, a − y)_{n}}{(a, a − x − y)_{n}}_{6}V_{5}(−a − n + x + y; 1 − a − b − n + x + y, x, y, −n; −1)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3238 \underline{Description}: Transformation formula ([1], Ex. 6, p. 97) in form of a rule. A q-analogue is HYPQ’s T5462.

3F_{2}[ \begin{matrix}\frac{1}{2} + \frac{a}{2}, 1 + \frac{a}{2}, −1 − a + b + c \\ b, c \end{matrix}; −\frac{4z}{(1 − z)^{2}} ] ⟶ \frac{(1 − z)^{1+a}}{(1 + z)}_{4}V_{3}(a; 1 + a − b, 1 + a − c; z)

Alphabetic List: T3263 119

T3239 \underline{Description}: Transformation formula ([1], Ex. 4.(iv), p. 97) in form of a rule. A q-analogue is HYPQ’s T3265. A

terminating q-analogue is HYPQ’s T3209.

3F_{2}[ \begin{matrix}a, b, c \\ 1 + a − b, 1 + a − c \end{matrix}; z] ⟶ (1 − z)^{−a}_{3}F_{2}[ \begin{matrix}\frac{a}{2}, \frac{1+a}{2}, 1 + a − b − c \\ 1 + a − b, 1 + a − c \end{matrix}; \frac{−4z}{(1 − z)^{2}} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3240 \underline{Description}: Transformation formula ([2], (3.5.10), q → 1, reversed) in form of a rule. A q-analogue is HYPQ’s T8711.

3F_{2}[ \begin{matrix}c, b, d \\ a, a − b + d \end{matrix}; 1] ⟶ Γ [ \begin{matrix}2a, 2a − 2b − c, a − b + d, a − c + d \\ 2a − 2b, 2a − c, a + d, a − b − c + d \end{matrix} ]_{7}V_{6}(−\frac{1}{2} + a; b, \frac{c}{2}, \frac{1}{2} + \frac{c}{2}, \frac{a}{2} − \frac{d}{2}, \frac{1}{2} + \frac{a}{2} − \frac{d}{2}; 1)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3261 \underline{Description}: Transformation formula ([2], (III.33), q ↑ 1) in form of a rule. A q-analogue is HYPQ’s T3261.

3F_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; 1] ⟶ Γ [ \begin{matrix}c, 1 + c − d, 1 − a, −b − c + e \\ −b + e, −c + e, 1 − a + c, 1 − d \end{matrix} ]_{3}F_{2}[ \begin{matrix}c, −a + d, 1 + c − e \\ 1 − a + c, 1 + b + c − e \end{matrix}; 1]

− Γ [ \begin{matrix}−1 + d, e, 1 + b − d, 1 + c − d, 1 − a, −b − c + e, 1 + b + c − e \\ 1 − d, 1 − d + e, b, c, −a + d, −1 − b − c + d + e, 2 + b + c − d − e \end{matrix} ]_{3}F_{2}[ \begin{matrix}1 + a − d, 1 + b − d, 1 + c − d \\ 2 − d, 1 − d + e \end{matrix}; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3262 \underline{Description}: Transformation formula ([7], (4.3.4.2)) in form of a rule. A q-analogue is HYPQ’s T3262.

3F_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; 1] ⟶ Γ [ \begin{matrix}e, −b − c + e \\ −b + e, −c + e \end{matrix} ]_{3}F_{2}[ \begin{matrix}−a + d, b, c \\ d, 1 + b + c − e \end{matrix}; 1]

+ Γ [ \begin{matrix}d, e, b + c − e, −a − b − c + d + e \\ −a + d, b, c, −b − c + d + e \end{matrix} ]_{3}F_{2}[ \begin{matrix}−b + e, −c + e, −a − b − c + d + e \\ −b − c + d + e, 1 − b − c + e \end{matrix}; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3263 \underline{Description}: Transformation formula ([2], Appendix (III.33), q → 1, reversed) in form of a rule. A q-analogue is

HYPQ’s T3263.

3F_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; 1] ⟶ Γ [ \begin{matrix}1 − c, d, −a − b + d, 1 + a − e \\ 1 + a − c, −a + d, −b + d, 1 − e \end{matrix} ]_{3}F_{2}[ \begin{matrix}1 + a − d, −c + e, a \\ 1 + a + b − d, 1 + a − c \end{matrix}; 1]

+ Γ [ \begin{matrix}1 − c, a + b − d, d, 1 + a − e, e, −a − b − c + d + e \\ a, b, 1 − b − c + d, 1 + a + b − d − e, −c + e, −a − b + d + e \end{matrix} ]_{3}F_{2}[ \begin{matrix}1 − b, −a − b − c + d + e, −b + d \\ 1 − a − b + d, 1 − b − c + d \end{matrix}; 1]

120 Alphabetic List: T4302

T3264 \underline{Description}: Transformation formula ([2], Appendix (III.34), q → 1, reversed) in form of a rule. A q-analogue is

HYPQ’s T3264.

3F_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; 1] ⟶ Γ [ \begin{matrix}1 + b − e, 1 + c − e \\ 1 − e, 1 + b + c − e \end{matrix} ]_{3}F_{2}[ \begin{matrix}−a + d, b, c \\ d, 1 + b + c − e \end{matrix}; 1]

− Γ [ \begin{matrix}d, 1 + a − e, 1 + b − e, 1 + c − e, −1 + e \\ a, b, c, 1 − e, 1 + d − e \end{matrix} ]_{3}F_{2}[ \begin{matrix}1 + c − e, 1 + b − e, 1 + a − e \\ 1 + d − e, 2 − e \end{matrix}; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3267 \underline{Description}: Transformation formula ([2], Ex. 3.6, q → 1) in form of a rule. A q-analogue is HYPQ’s T3267.

3F_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; 1] ⟶ Γ [ \begin{matrix}d, 1 + a − e, 1 + b − e, 1 + c − e \\ −a + d, 1 − e, 1 + a + b − e, 1 + a + c − e \end{matrix} ]_{3}F_{2}[ \begin{matrix}a, 1 + a − e, 1 + a + b + c − d − e \\ 1 + a + b − e, 1 + a + c − e \end{matrix}; 1]

− Γ [ \begin{matrix}d, 1 + a − e, 1 + b − e, 1 + c − e, −1 + e \\ a, b, c, 1 − e, 1 + d − e \end{matrix} ]_{3}F_{2}[ \begin{matrix}1 + a − e, 1 + b − e, 1 + c − e \\ 2 − e, 1 + d − e \end{matrix}; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T3268 \underline{Description}: Transformation formula ([2], Ex. 3.6, q → 1, reversed) in form of a rule. A q-analogue is HYPQ’s T3268.

3F_{2}[ \begin{matrix}a, b, c \\ d, e \end{matrix}; 1] ⟶ Γ [ \begin{matrix}a − b, d, e, −a − b − c + d + e \\ a, −b + d, −b + e, −a − c + d + e \end{matrix} ]_{3}F_{2}[ \begin{matrix}b, −a + d, −a + e \\ 1 − a + b, −a − c + d + e \end{matrix}; 1]

+ Γ [ \begin{matrix}−a + b, d, e, −a − b − c + d + e \\ b, −a + d, −a + e, −b − c + d + e \end{matrix} ]_{3}F_{2}[ \begin{matrix}a, −b + d, −b + e \\ −b − c + d + e, 1 + a − b \end{matrix}; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4301 \underline{Description}: Transformation formula ([7], (4.3.5.1)) in form of a rule. A q-analogue is HYPQ’s T4301.

4F_{3}[ \begin{matrix}a, b, c, −n \\ e, f, 1 + a + b + c − e − f − n \end{matrix}; 1]

⟶ \frac{(−a + e, −a + f)_{n}}{(e, f)_{n}}_{4}F_{3}[ \begin{matrix}−n, a, 1 + a + c − e − f − n, 1 + a + b − e − f − n \\ 1 + a + b + c − e − f − n, 1 + a − e − n, 1 + a − f − n \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4302 \underline{Description}: Transformation formula ([2], Appendix (III.16), q ↑ 1) in form of a rule. A q-analogue is HYPQ’s T4302.

4F_{3}[ \begin{matrix}a, b, c, −n \\ e, f, 1 + a + b + c − e − f − n \end{matrix}; 1]

⟶ \frac{(a, −a − b + e + f, −a − c + e + f)_{n}}{(e, f, −a − b − c + e + f)_{n}}_{4}F_{3}[ \begin{matrix}−n, −a + e, −a + f, −a − b − c + e + f \\ −a − b + e + f, −a − c + e + f, 1 − a − n \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

Alphabetic List: T4310 121

T4303 \underline{Description}: Transformation formula ([7], (2.4.1.1)) in form of a rule. A q-analogue is HYPQ’s T4303.

4F_{3}[ \begin{matrix}a, b, c, −n \\ e, f, 1 + a + b + c − e − f − n \end{matrix}; 1]

⟶ \frac{(−a − b + e + f, −a − c + e + f)_{n}}{(−a + e + f, −a − b − c + e + f)_{n}}_{7}V_{6}(−1 − a + e + f; −a + f, −a + e, b, c, −n; 1)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4304 \underline{Description}: Transformation formula ([7], (4.3.6.4)) in form of a rule. A q-analogue is HYPQ’s T4304.

4F_{3}[ \begin{matrix}a, b, c, −n \\ e, f, 1 + a + b + c − e − f − n \end{matrix}; 1] ⟶ Γ [ \begin{matrix}−b − c + e + f + n, −a − c + e + f + n, \\ −c + e + f + n, −b + e + f + n, \end{matrix}

\begin{matrix}−a − b + e + f + n, e + f + n \\ −a + e + f + n, −a − b − c + e + f + n \end{matrix} ]_{7}V_{6}(−1 + e + f + n; a, b, c, e + n, f + n; 1)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4306 \underline{Description}: Transformation formula ([2], Appendix (III.21), q ↑ 1, reversed) in form of a rule. A q-analogue is

HYPQ’s T4306.

4F_{3}[ \begin{matrix}a, b, c, d \\ \frac{1}{2} + a + b, \frac{c+d}{2}, \frac{1+c+d}{2} \end{matrix} ; 1] ⟶ _{3}F_{2}[ \begin{matrix}2a, 2b, c \\ \frac{1}{2} + a + b, c + d \end{matrix}; 1]

provided both series terminate.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4309 \underline{Description}: Transformation formula ([2], Ex. 2.13(i), q → 1) in form of a rule. A q-analogue is HYPQ’s T4309.

4F_{3}[ \begin{matrix}a, b, c, d \\ 1 + a − b, 1 + a − c, 1 + a − d \end{matrix}; 1] ⟶ Γ [ \begin{matrix}3 + 2a − 2b − 2c − 2d, 2 + 2a − b − c − d \\ 3 + 3a − 2b − 2c − 2d, 2 + a − b − c − d \end{matrix} ]

7V_{6}(1 + 2a − b − c − d; \frac{a}{2}, \frac{1}{2} + \frac{a}{2}, 1 + a − c − d, 1 + a − b − d, 1 + a − b − c; 1)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4310 \underline{Description}: Transformation formula ([2], Ex. 2.13(ii), q → 1) in form of a rule. A q-analogue is HYPQ’s T4310.

4F_{3}[ \begin{matrix}a, b, c, d \\ 1 + a − b, 1 + a − c, 1 + a − d \end{matrix}; −1]

⟶ Γ [ \begin{matrix}2 + 2a − b − c − d, 1 + \frac{a}{2} \\ 1 + a, 2 + \frac{3a}{2} − b − c − d \end{matrix} ]_{6}V_{5}(1 + 2a − b − c − d; \frac{a}{2}, 1 + a − c − d, 1 + a − b − d, 1 + a − b − c; −1)

T4312 \underline{Description}: Transformation formula ([2], Ex. 3.4, q → 1) in form of a rule. A q-analogue is HYPQ’s T4312.

4F_{3}[ \begin{matrix}\frac{a}{2}, \frac{1}{2} + \frac{a}{2}, \frac{1}{2} − \frac{n}{2}, −\frac{n}{2} \\ \frac{d}{2}, \frac{1}{2} + \frac{d}{2}, \frac{1}{2} + b \end{matrix} ; 1] ⟶ \frac{(d − a)_{n}}{(d)_{n}}_{3}F_{2}[ \begin{matrix}a, b, −n \\ 1 + a − d − n, 2b \end{matrix}; 2]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4331 \underline{Description}: Transformation formula ([1], Ex. 6, p. 97, reversed) in form of a rule. A q-analogue is HYPQ’s T5461.

4V_{3}(a; b, c; z) ⟶ \frac{(1 + z)}{(1 − z)^{a+1}}_{3}F_{2}[ \begin{matrix}\frac{1}{2} + \frac{a}{2}, 1 + \frac{a}{2}, 1 + a − b − c \\ 1 + a − b, 1 + a − c \end{matrix}; −\frac{4z}{(1 − z)^{2}} ]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4332 \underline{Description}: Transformation formula ([1], 4.6(1)) in form of a rule. A q-analogue is HYPQ’s T10904.

4F_{3}[ \begin{matrix}b, x, y, −n \\ a − x, a − y, a + n \end{matrix}; 1] ⟶ \frac{(a, a − x − y)_{n}}{(a − x, a − y)_{n}}_{5}F_{4}[ \begin{matrix}x, y, \frac{a}{2} − \frac{b}{2}, \frac{1}{2} + \frac{a}{2} − \frac{b}{2}, −n \\ a − b, \frac{a}{2}, \frac{1}{2} + \frac{a}{2}, 1 − a − n + x + y \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T4362 \underline{Description}: Transformation formula ([7], (2.4.4.3), reversed) in form of a rule. A q-analogue is HYPQ’s T4362.

4F_{3}[ \begin{matrix}a, b, c, d \\ e, f, 1 + a + b + c + d − e − f \end{matrix} ; 1]

⟶ Γ [ \begin{matrix}−a + e + f, −a − b − c + e + f, −a − b − d + e + f, −a − c − d + e + f \\ −a − b + e + f, −a − c + e + f, −a − d + e + f, −a − b − c − d + e + f \end{matrix} ]

7V_{6}(−1 − a + e + f; −a + f, −a + e, b, c, d; 1)

− Γ [ \begin{matrix}e, a + b + c + d − e − f, f, −a − b − c + e + f, −a − b − d + e + f, −a − c − d + e + f, −b − c − d + e + f \\ a, b, c, d, −a − b − c − d + e + f, −a − b − c − d + 2e + f, −a − b − c − d + e + 2f \end{matrix} ]

4F_{3}[ \begin{matrix}−a − b − c + e + f, −a − b − d + e + f, −a − c − d + e + f, −b − c − d + e + f \\ −a − b − c − d + 2e + f, −a − b − c − d + e + 2f, 1 − a − b − c − d + e + f \end{matrix} ; 1]

Alphabetic List: T5403 123

T4391 \underline{Description}: Transformation formula ([2], (3.5.7), q → 1, reversed) in form of a rule. A q-analogue is HYPQ’s T5465.

4F_{3}[ \begin{matrix}a, c, d, e \\ b, 2a, 1 − 2b + c + d + e \end{matrix}; 1]

⟶ Γ [ \begin{matrix}b, \frac{1}{2} + b, b − \frac{c}{2} − \frac{d}{2}, \frac{1}{2} + b − \frac{c}{2} − \frac{d}{2}, b − \frac{c}{2} − \frac{e}{2}, \frac{1}{2} + b − \frac{c}{2} − \frac{e}{2}, b − \frac{d}{2} − \frac{e}{2}, \frac{1}{2} + b − \frac{d}{2} − \frac{e}{2} \\ b − \frac{c}{2}, \frac{1}{2} + b − \frac{c}{2}, b − \frac{d}{2}, \frac{1}{2} + b − \frac{d}{2}, b − \frac{e}{2}, \frac{1}{2} + b − \frac{e}{2}, b − \frac{c}{2} − \frac{d}{2} − \frac{e}{2}, \frac{1}{2} + b − \frac{c}{2} − \frac{d}{2} − \frac{e}{2} \end{matrix} ]

9V_{8}(−\frac{1}{2} + b; −a + b, \frac{c}{2}, \frac{1}{2} + \frac{c}{2}, \frac{d}{2}, \frac{1}{2} + \frac{d}{2}, \frac{e}{2}, \frac{1}{2} + \frac{e}{2}; 1)

−Γ [ \begin{matrix}\frac{1}{2} + a, b, b − \frac{c}{2} − \frac{d}{2}, \frac{1}{2} + b − \frac{c}{2} − \frac{d}{2}, a + 2b − c − d − e, b − \frac{c}{2} − \frac{e}{2}, \frac{1}{2} + b − \frac{c}{2} − \frac{e}{2}, b − \frac{d}{2} − \frac{e}{2}, \frac{1}{2} + b − \frac{d}{2} − \frac{e}{2}, \\ \frac{c}{2}, \frac{1}{2} + \frac{c}{2}, \frac{d}{2}, \frac{1}{2} + \frac{d}{2}, 3b − c − d − e, b − \frac{c}{2} − \frac{d}{2} − \frac{e}{2}, \frac{1}{2} + b − \frac{c}{2} − \frac{d}{2} − \frac{e}{2}, a + b − \frac{c}{2} − \frac{d}{2} − \frac{e}{2}, \frac{1}{2} + a + b − \frac{c}{2} − \frac{d}{2} − \frac{e}{2}, \end{matrix}

\begin{matrix}−b + \frac{c}{2} + \frac{d}{2} + \frac{e}{2}, \frac{1}{2} − b + \frac{c}{2} + \frac{d}{2} + \frac{e}{2} \\ \frac{e}{2}, \frac{1}{2} + \frac{e}{2} \end{matrix} ]_{4}F_{3}[ \begin{matrix}a + 2b − c − d − e, 2b − c − d, 2b − c − e, 2b − d − e \\ 3b − c − d − e, 2a + 2b − c − d − e, 1 + 2b − c − d − e \end{matrix}; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5401 \underline{Description}: Transformation formula ([7], (2.4.3.4)) in form of a rule. A q-analogue is HYPQ’s T5401.

5F_{4}[ \begin{matrix}a, b, c, d, −n \\ 1 + a − b, 1 + a − c, 1 + a − d, −2 − 2a + 2b + 2c + 2d − n \end{matrix}; 1]

⟶ \frac{(2 + a − b − c − d, 3 + 3a − 2b − 2c − 2d)_{n}}{(2 + 2a − b − c − d, 3 + 2a − 2b − 2c − 2d)_{n}}_{9}V_{8}(1 + 2a − b − c − d; 1 + a − c − d,

1 + a − b − d, 1 + a − b − c, \frac{a}{2}, \frac{1}{2} + \frac{a}{2}, 3 + 3a − 2b − 2c − 2d + n, −n; 1)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5402 \underline{Description}: Transformation formula ([2], Appendix (III.26), q ↑ 1) in form of a rule. A q-analogue is HYPQ’s T5402.

5F_{4}[ \begin{matrix}−n, b, c, d, e \\ 1 − b − n, 1 − c − n, 1 − d − n, −2 + 2b + 2c + 2d + e + 2n \end{matrix}; 1]

⟶ \frac{(2 − b − c − d − e − 2n, 3 − 2b − 2c − 2d − 3n)_{n}}{(2 − b − c − d − 2n, 3 − 2b − 2c − 2d − e − 3n)_{n}}_{9}V_{8}(1 − b − c − d − 2n; 1 − c − d − n,

1 − b − d − n, 1 − b − c − n, \frac{−n}{2}, \frac{1}{2} − \frac{n}{2}, e, 3 − 2b − 2c − 2d − e − 3n; 1)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T5403 \underline{Description}: Transformation formula ([1], 4.6(1), reversed) in form of a rule. A q-analogue is HYPQ’s T5403.

5F_{4}[ \begin{matrix}x, y, a, \frac{1}{2} + a, −n \\ 2a, b, \frac{1}{2} + b, 1 − 2b − n + x + y \end{matrix}; 1] ⟶ \frac{(2b − x, 2b − y)_{n}}{(2b, 2b − x − y)_{n}}_{4}F_{3}[ \begin{matrix}−2a + 2b, x, y, −n \\ 2b − x, 2b − y, 2b + n \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

5F_{4}[ \begin{matrix}a, b, c, d, e \\ 1 + a − b, 1 + a − c, 1 + a − d, −2 − 2a + 2b + 2c + 2d + e \end{matrix}; 1]

+ Γ [ \begin{matrix}1 + a − b, 1 + a − c, 3 + 2a − 2b − 2c − 2d, 1 + a − d, 3 + 3a − 2b − 2c − 2d − e, 3 + 2a − b − 2c − 2d − e, \\ a, b, c, d, 4 + 3a − 2b − 2c − 3d − e, 4 + 3a − 2b − 3c − 2d − e, \end{matrix} ]

\begin{matrix}3 + 2a − 2b − c − 2d − e, 3 + 2a − 2b − 2c − d − e, −3 − 2a + 2b + 2c + 2d + e \\ 4 + 3a − 3b − 2c − 2d − e, 3 + 2a − 2b − 2c − 2d − e, e \end{matrix}

5F_{4}[ \begin{matrix}3 + 2a − 2b − 2c − 2d, 3 + 3a − 2b − 2c − 2d − e, 3 + 2a − b − 2c − 2d − e, \\ 4 + 2a − 2b − 2c − 2d − e, 4 + 3a − 3b − 2c − 2d − e, \end{matrix}

\begin{matrix}3 + 2a − 2b − c − 2d − e, 3 + 2a − 2b − 2c − d − e \\ 4 + 3a − 2b − 3c − 2d − e, 4 + 3a − 2b − 2c − 3d − e \end{matrix}; 1]

⟶ Γ [ \begin{matrix}3 + 2a − 2b − 2c − 2d, 2 + 2a − b − c − d, 3 + 3a − 2b − 2c − 2d − e, 2 + a − b − c − d − e \\ 3 + 3a − 2b − 2c − 2d, 2 + a − b − c − d, 3 + 2a − 2b − 2c − 2d − e, 2 + 2a − b − c − d − e \end{matrix} ]

9V_{8}(1 + 2a − b − c − d; \frac{a}{2}, \frac{1}{2} + \frac{a}{2}, 1 + a − c − d, 1 + a − b − d, 1 + a − b − c, e, 3 + 3a − 2b − 2c − 2d − e; 1)

+ Γ [ \begin{matrix}1 + a − b, 1 + a − c, 3 + 2a − 2b − 2c − 2d, 1 + a − d, 6 + 4a − 3b − 3c − 3d − 2e, 3 + 3a − 2b − 2c − 2d − e, \\ a, 1 + a − b − c, 1 + a − b − d, 1 + a − c − d, 7 + 5a − 4b − 4c − 4d − 2e, \end{matrix} ]

\begin{matrix}3 + 2a − b − 2c − 2d − e, 3 + 2a − 2b − c − 2d − e, 3 + 2a − 2b − 2c − d − e, −2 − a + b + c + d + e \\ 3 + 2a − 2b − 2c − 2d − e, 3 + 2a − b − c − 2d − e, 3 + 2a − b − 2c − d − e, 3 + 2a − 2b − c − d − e, e \end{matrix} ]

9V_{8}(5 + 4a − 3b − 3c − 3d − 2e; 2 + \frac{3a}{2} − b − c − d − e, \frac{5}{2} + \frac{3a}{2} − b − c − d − e, 2 + a − b − c − d,

3 + 3a − 2b − 2c − 2d − e, 3 + 2a − b − 2c − 2d − e, 3 + 2a − 2b − c − 2d − e, 3 + 2a − 2b − 2c − d − e; 1)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T6501 \underline{Description}: Transformation formula ([7], (2.4.3.3)) in form of a rule. A q-analogue is HYPQ’s T6501.

6F_{5}[ \begin{matrix}a, 1 + \frac{a}{2}, b, c, d, −n \\ \frac{a}{2}, 1 + a − b, 1 + a − c, 1 + a − d, −1 − 2a + 2b + 2c + 2d − n \end{matrix}; 1]

⟶ \frac{(2 + 3a − 2b − 2c − 2d, 1 + a − b − c − d)_{n}}{(2 + 2a − 2b − 2c − 2d, 2 + 2a − b − c − d)_{n}}

9V_{8}(1 + 2a − b − c − d; 1 + a − c − d, 1 + a − b − d, 1 + a − b − c, 1 + \frac{a}{2}, \frac{1}{2} + \frac{a}{2}, 2 + 3a − 2b − 2c − 2d + n, −n; 1)

Alphabetic List: T7631 125

T6531 \underline{Description}: Transformation formula ([7], (2.4.3.5)) in form of a rule. A q-analogue is HYPQ’s T7601.

6F_{5}[ \begin{matrix}a, 1 + \frac{a}{2}, b, c, d, −n \\ \frac{a}{2}, 1 + a − b, 1 + a − c, 1 + a − d, −2a + 2b + 2c + 2d − n \end{matrix}; 1]

⟶ \frac{(1 + 3a − 2b − 2c − 2d + 2n)}{(1 + 3a − 2b − 2c − 2d)} \frac{(a − b − c − d, 1 + 3a − 2b − 2c − 2d)_{n}}{(2 + 2a − b − c − d, 1 + 2a − 2b − 2c − 2d)_{n}}_{9}V_{8}(1 + 2a − b − c − d;

1 + a − c − d, 1 + a − b − d, 1 + a − b − c, \frac{1}{2} + \frac{a}{2}, 1 + \frac{a}{2}, 1 + 3a − 2b − 2c − 2d + n, −n; 1)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T6532 \underline{Description}: Transformation formula ([2], Ex. 2.13(ii), q → 1, reversed) in form of a rule. A q-analogue is HYPQ’s

  T8706.

6V_{5}(a; b, c, d, 1 + 2a − 2b − c − d; −1) ⟶ Γ [ \begin{matrix}1 + 2b, 1 + a − b \\ 1 + b, 1 + a \end{matrix} ]_{4}F_{3}[ \begin{matrix}2b, −a + 2b + c, −a + 2b + d, 1 + a − c − d \\ 1 + a − c, 1 + a − d, −a + 2b + c + d \end{matrix}; −1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T6533 \underline{Description}: Transformation formula ([2], (3.10.4), q → 1) in form of a rule. A q-analogue is HYPQ’s T10904.

6V_{5}(a; b, x, y, −n; −1) ⟶ \frac{(1 + a, 1 + a − x − y)_{n}}{(1 + a − x, 1 + a − y)_{n}}_{3}F_{2}[ \begin{matrix}−n, x, y \\ −a − n + x + y, 1 + a − b \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T6534 \underline{Description}: Transformation formula ([1], 4.4(2)) in form of a rule. A q-analogue is HYPQ’s T7701.

6V_{5}(a; b, c, d, e; −1) ⟶ Γ [ \begin{matrix}1 + a − d, 1 + a − e \\ 1 + a, 1 + a − d − e \end{matrix} ]_{3}F_{2}[ \begin{matrix}1 + a − b − c, d, e \\ 1 + a − b, 1 + a − c \end{matrix}; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T7631 \underline{Description}: Transformation formula ([7], (2.4.1.1), reversed) in form of a rule. A q-analogue is HYPQ’s T8701.

7V_{6}(a; b, c, d, e, f; 1) ⟶ Γ [ \begin{matrix}1 + a − d, 1 + a − e, 1 + a − f, 1 + a − d − e − f \\ 1 + a, 1 + a − d − e, 1 + a − d − f, 1 + a − e − f \end{matrix} ]

4F_{3}[ \begin{matrix}1 + a − b − c, d, e, f \\ 1 + a − b, 1 + a − c, −a + d + e + f \end{matrix} ; 1]

provided the _{7}F_{6} series converges and the _{4}F_{3} series terminates.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

126 Alphabetic List: T7636

T7632 \underline{Description}: Transformation formula ([7], (2.4.1.1), reversed) in form of a rule. A q-analogue is HYPQ’s T8702.

7V_{6}(a; b, c, d, e, −n; 1) ⟶ \frac{(1 + a, 1 + a − d − e)_{n}}{(1 + a − d, 1 + a − e)_{n}}_{4}F_{3}[ \begin{matrix}1 + a − b − c, d, e, −n \\ 1 + a − b, 1 + a − c, −a + d + e − n \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T7633 \underline{Description}: Transformation formula ([7], (4.3.6.4), reversed) in form of a rule. A q-analogue is HYPQ’s T8703.

7V_{6}(a; b, c, d, e, 1 + a − e + n; 1)

⟶ Γ [ \begin{matrix}1 + a − d, 1 + a − c, 1 + a − b, 1 + a − b − c − d \\ 1 + a − c − d, 1 + a − b − d, 1 + a − b − c, 1 + a \end{matrix} ]_{4}F_{3}[ \begin{matrix}b, c, d, −n \\ 1 + a − e, −a + b + c + d, e − n \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T7634 \underline{Description}: Transformation formula ([1], 7.5.(1)) in form of a rule. A q-analogue is HYPQ’s T8704.

7V_{6}(a; b, c, d, e, f; 1) ⟶ Γ [ \begin{matrix}1 + a − e, 1 + a − f, 2 + 2a − b − c − d, 2 + 2a − b − c − d − e − f \\ 1 + a, 1 + a − e − f, 2 + 2a − b − c − d − e, 2 + 2a − b − c − d − f \end{matrix} ]

7V_{6}(1 + 2a − b − c − d; 1 + a − c − d, 1 + a − b − d, 1 + a − b − c, e, f; 1)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T7635 \underline{Description}: Transformation formula ([1], 7.5.(2)) in form of a rule. A q-analogue is HYPQ’s T8705.

7V_{6}(a; b, c, d, e, f; 1) ⟶ Γ [ \begin{matrix}1 + a − c, 1 + a − d, 1 + a − e, 1 + a − f, 3 + 3a − 2b − c − d − e − f, \\ 1 + a, b, 2 + 2a − b − d − e − f, 2 + 2a − b − c − e − f, 2 + 2a − b − c − d − f, \end{matrix}

\begin{matrix}2 + 2a − b − c − d − e − f \\ 2 + 2a − b − c − d − e \end{matrix} ]_{7}V_{6}(2 + 3a − 2b − c − d − e − f; 1 + a − b − c, 1 + a − b − d,

1 + a − b − e, 1 + a − b − f, 2 + 2a − b − c − d − e − f; 1)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T7636 \underline{Description}: Transformation formula ([2], (3.5.10), q → 1) in form of a rule. A q-analogue is HYPQ’s T8710.

7V_{6}(a; b, c, \frac{1}{2} + c, d, \frac{1}{2} + d; 1)

⟶ Γ [ \begin{matrix}1 + 2a − 2b, 1 + 2a − 2c, 1 + 2a − 2d, 1 + 2a − b − 2c − 2d \\ 1 + 2a, 1 + 2a − 2b − 2c, 1 + 2a − b − 2d, 1 + 2a − 2c − 2d \end{matrix} ]_{3}F_{2}[ \begin{matrix}2c, b, \frac{1}{2} + a − 2d \\ 1 + 2a − b − 2d, \frac{1}{2} + a \end{matrix}; 1]

Alphabetic List: T7692 127

T7637 \underline{Description}: Transformation formula ([2], Ex. 2.13(i), q → 1, reversed) in form of a rule. A q-analogue is HYPQ’s

  T10902.

7V_{6}(a; b, \frac{1}{2} + b, c, d, 1 + 2a − 2b − c − d; 1)

⟶ Γ [ \begin{matrix}1 + a − 2b, 1 + 2a − 2b \\ 1 + a, 1 + 2a − 4b \end{matrix} ]_{4}F_{3}[ \begin{matrix}2b, −a + 2b + c, −a + 2b + d, 1 + a − c − d \\ 1 + a − c, 1 + a − d, −a + 2b + c + d \end{matrix}; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T7691 \underline{Description}: Transformation formula ([7], (2.4.4.3)) in form of a rule. A q-analogue is HYPQ’s T8761.

7V_{6}(a; b, c, d, e, f; 1)

⟶ Γ [ \begin{matrix}1 + a − d, 1 + a − e, 1 + a − f, 1 + a − d − e − f \\ 1 + a, 1 + a − d − e, 1 + a − d − f, 1 + a − e − f \end{matrix} ]_{4}F_{3}[ \begin{matrix}1 + a − b − c, d, e, f \\ 1 + a − b, 1 + a − c, −a + d + e + f \end{matrix} ; 1]

+ Γ [ \begin{matrix}1 + a − b, 1 + a − c, 1 + a − d, 1 + a − e, 1 + a − f, 2 + 2a − b − c − d − e − f, −1 − a + d + e + f \\ 1 + a, 1 + a − b − c, d, e, f, 2 + 2a − b − d − e − f, 2 + 2a − c − d − e − f \end{matrix} ]

4F_{3}[ \begin{matrix}1 + a − d − e, 1 + a − d − f, 1 + a − e − f, 2 + 2a − b − c − d − e − f \\ 2 + 2a − b − d − e − f, 2 + 2a − c − d − e − f, 2 + a − d − e − f \end{matrix} ; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T7692 \underline{Description}: Transformation formula ([7], (4.3.7.8)) in form of a rule. A q-analogue is HYPQ’s T8762.

7V_{6}(a;b, c, d, e, f; 1)

⟶ −Γ [ \begin{matrix}a − b, 1 + a − c, 1 + a − d, 1 + a − e, 1 + a − f, −a + b + d, −a + b + e, −a + b + f, −a + d + e + f, \\ 1 + a, −a + b, 1 + b − c, 1 + b − d, 1 + b − e, 1 + b − f, d, e, f, 1 + a − b − c, −2a + b + d + e + f, \end{matrix}

\begin{matrix}a + 1 − d − e − f, 1 − c, 1 − a + 2b \\ 1 + 2a − b − d − e − f \end{matrix} ]_{7}V_{6}(−a + 2b; b, −a + b + c, −a + b + d, −a + b + e, −a + b + f; 1)

+ Γ [ \begin{matrix}1 + a − d, 1 + a − e, 1 + a − f, 1 + a − d − e − f, 1 − c, 1 − c + e + f, −a + b + e, −a + b + f \\ 1 + a, 1 + a − d − e, 1 + a − d − f, 1 + a − e − f, 1 − c + e, 1 − c + f, −a + b + e + f \end{matrix} ]

7V_{6}(−c + e + f; 1 + a − b − c, 1 + a − c − d, −a + e + f, e, f; 1)

128 Alphabetic List: T8731

T7693 \underline{Description}: Transformation formula ([2], Ex. 2.15, q → 1) in form of a rule. A q-analogue is HYPQ’s T8764.

7V_{6}(2a; a + b, a + c, a + d, a + e, a + f; 1)

⟶ −Γ [ \begin{matrix}a − b, 1 + 2b, −b + c, b + c, 1 − a − d, 1 + a − d, 1 − a − e, 1 + a − e, 1 − a − f, 1 + a − f \\ 1 + 2a, −a + b, −a + c, a + c, 1 − b − d, 1 + b − d, 1 − b − e, 1 + b − e, 1 − b − f, 1 + b − f \end{matrix} ]

7V_{6}(2b; a + b, b + c, b + d, b + e, b + f; 1)

− Γ [ \begin{matrix}a − c, b − c, b + c, 1 + 2c, 1 − a − d, 1 + a − d, 1 − a − e, 1 + a − e, 1 − a − f, 1 + a − f \\ 1 + 2a, −a + b, a + b, −a + c, 1 − c − d, 1 + c − d, 1 − c − e, 1 + c − e, 1 − c − f, 1 + c − f \end{matrix} ]

7V_{6}(2c; a + c, b + c, c + d, c + e, c + f; 1)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T7694 \underline{Description}: Transformation formula ([7], (4.3.7.8), reversed) in form of a rule. A q-analogue is HYPQ’s T8763.

7V_{6}(a; b, c, d, e, f; 1)

⟶ Γ [ \begin{matrix}1 + a − b, 1 − d, 1 + a − e, −a + c + e, 1 + a − f, 1 + a − b − e − f, −a + c + f, 1 − d + e + f \\ 1 + a, −a + c, 1 + a − b − e, 1 − d + e, 1 + a − b − f, 1 + a − e − f, 1 − d + f, −a + c + e + f \end{matrix} ]

7V_{6}(−d + e + f; 1 + a − b − d, −a + e + f, 1 + a − c − d, e, f; 1)

+ Γ [ \begin{matrix}1 + a − b, 1 + a − c, 1 − d, 1 + a − d, 1 + a − e, −a + c + e, 1 + a − f, 3 + 2a − 2b − d − e − f, \\ 1 + a, b, 1 − b + c, 1 + a − c − d, 2 + a − b − d − e, e, 2 + a − b − d − f, \end{matrix}

\begin{matrix}2 + 2a − b − c − d − e − f, −a + c + f, −1 − a + b + e + f \\ 2 + 2a − b − c − e − f, 2 + 2a − b − d − e − f, f, −1 − 2a + b + c + e + f \end{matrix} ]

7V_{6}(2 + 2a − 2b − d − e − f; 1 + a − b − d, 1 − b, 2 + 2a − b − c − d − e − f, 1 + a − b − f, 1 + a − b − e; 1)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T8731 \underline{Description}: Transformation formula ([6], (7.7), q → 1, reversed) in form of a rule. A q-analogue is HYPQ’s T10906.

8V_{7}(2a + n; c, d, e, \frac{1}{2} + a + n, 1 + 4a − c − d − e + n, −n; −1)

⟶ \frac{(1 + 2a − c, 1 + 2a − d, 1 + 2a − e, 1 + 2a − c − d − e)_{n}}{(1 + 2a, 1 + 2a − c − d, 1 + 2a − c − e, 1 + 2a − d − e)_{n}}

11V_{10}(a; −n, \frac{c}{2}, \frac{1}{2} + \frac{c}{2}, \frac{d}{2}, \frac{1}{2} + \frac{d}{2}, \frac{e}{2}, \frac{1}{2} + \frac{e}{2}, \frac{1}{2} + 2a − \frac{c}{2} − \frac{d}{2} − \frac{e}{2} + \frac{n}{2}, 1 + 2a − \frac{c}{2} − \frac{d}{2} − \frac{e}{2} + \frac{n}{2}; 1)

T8732 \underline{Description}: Transformation formula ([6], (7.8), q → 1, reversed) in form of a rule. A q-analogue is HYPQ’s T10907.

8V_{7}(2a − e; \frac{1}{2} + a − e, c, d, e, 1 + 4a − c − d − e + n, −n; −1)

⟶ \frac{(1 + 2a − c, 1 + 2a − d, 1 + 2a − e, 1 + 2a − c − d − e)_{n}}{(1 + 2a, 1 + 2a − c − d, 1 + 2a − c − e, 1 + 2a − d − e)_{n}}

11V_{10}(a; e, \frac{c}{2}, \frac{1}{2} + \frac{c}{2}, \frac{d}{2}, \frac{1}{2} + \frac{d}{2}, \frac{1}{2} + 2a − \frac{c}{2} − \frac{d}{2} − \frac{e}{2} + \frac{n}{2}, 1 + 2a − \frac{c}{2} − \frac{d}{2} − \frac{e}{2} + \frac{n}{2}, \frac{1}{2} − \frac{n}{2}, \frac{−n}{2}; 1)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T9831 \underline{Description}: Transformation formula ([7], (2.4.4.1)) in form of a rule. A q-analogue is HYPQ’s T10901.

9V_{8}(a; b, c, d, e, f, 2 + 3a − b − c − d − e − f + n, −n; 1)

⟶ \frac{(1 + a, 1 + a − e − f, 2 + 2a − b − c − d − e, 2 + 2a − b − c − d − f)_{n}}{(1 + a − e, 1 + a − f, 2 + 2a − b − c − d − e − f, 2 + 2a − b − c − d)_{n}}

9V_{8}(1 + 2a − b − c − d; 1 + a − c − d, 1 + a − b − d, 1 + a − b − c, e, f, 2 + 3a − b − c − d − e − f + n, −n; 1)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T9832 \underline{Description}: Transformation formula ([7], (2.4.3.4), reversed) in form of a rule. A q-analogue is HYPQ’s T121101.

9V_{8}(a; b, c, d, \frac{1}{2} + a − \frac{b + c + d}{2}, 1 + a − \frac{b + c + d}{2}, b + c + d + n, −n; 1)

⟶ \frac{(1 + a, −1 − 2a + 2b + 2c + 2d)_{n}}{(−a + b + c + d, b + c + d)_{n}}_{5}F_{4}[ \begin{matrix}1 + 2a − b − c − d, 1 + a − c − d, 1 + a − b − d, 1 + a − b − c, −n \\ 1 + a − b, 1 + a − c, 1 + a − d, 2 + 2a − 2b − 2c − 2d − n \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T9833 \underline{Description}: Transformation formula ([2], Appendix (III.26), q ↑ 1, reversed) in form of a rule. A q-analogue is

  HYPQ’s T121102.

9V_{8}(\frac{−1 + b + c + d − n}{2}; b, c, d, \frac{−n}{2}, \frac{1}{2} − \frac{n}{2}, e, b + c + d − e; 1) ⟶ \frac{(\frac{1}{2} + \frac{b}{2} + \frac{c}{2} + \frac{d}{2} − \frac{n}{2}, b + c + d − e)_{n}}{(\frac{1}{2} + \frac{b}{2} + \frac{c}{2} + \frac{d}{2} − e − \frac{n}{2}, b + c + d)_{n}}

5F_{4}[ \begin{matrix}−n, \frac{1}{2} + \frac{b}{2} − \frac{c}{2} − \frac{d}{2} − \frac{n}{2}, \frac{1}{2} − \frac{b}{2} + \frac{c}{2} − \frac{d}{2} − \frac{n}{2}, \frac{1}{2} − \frac{b}{2} − \frac{c}{2} + \frac{d}{2} − \frac{n}{2}, e \\ \frac{1}{2} − \frac{b}{2} + \frac{c}{2} + \frac{d}{2} − \frac{n}{2}, \frac{1}{2} + \frac{b}{2} − \frac{c}{2} + \frac{d}{2} − \frac{n}{2}, \frac{1}{2} + \frac{b}{2} + \frac{c}{2} − \frac{d}{2} − \frac{n}{2}, 1 − b − c − d + e − n \end{matrix}; 1]

T9834 \underline{Description}: Transformation formula ([7], (2.4.3.5), reversed) in form of a rule. A q-analogue is HYPQ’s T121103.

9V_{8}(a;b, c, d, 1 + a − \frac{b + c + d}{2}, \frac{3}{2} + a − \frac{b + c + d}{2}, −2 + b + c + d + n, −n; 1)

⟶ \frac{(−2 + b + c + d)}{(−2 + b + c + d + 2n)} \frac{(1 + a, −3 − 2a + 2b + 2c + 2d)_{n}}{(−2 − a + b + c + d, −2 + b + c + d)_{n}}

6F_{5}[ \begin{matrix}1 + 2a − b − c − d, \frac{3}{2} + a − \frac{b}{2} − \frac{c}{2} − \frac{d}{2}, 1 + a − c − d, 1 + a − b − d, 1 + a − b − c, −n \\ \frac{1}{2} + a − \frac{b}{2} − \frac{c}{2} − \frac{d}{2}, 1 + a − b, 1 + a − c, 1 + a − d, 4 + 2a − 2b − 2c − 2d − n \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T9835 \underline{Description}: Transformation formula ([1], 7.6(1)) in form of a rule. A q-analogue is HYPQ’s T10903.

9V_{8}(a; b, c, d, e, f, 2 + 3a − b − c − d − e − f + n, −n; 1)

⟶ \frac{(1 + a, b, 1 + a − c − e, 1 + a − d − e, 1 + a − e − f, −1 − 2a + b + c + d + f − n)_{n}}{(1 + a − c, 1 + a − d, 1 + a − e, b − e, 1 + a − f, −1 − 2a + b + c + d + e + f − n)_{n}}

9V_{8}(−b + e − n; e, 1 + a − b − c, 1 + a − b − d, 1 + a − b − f, −1 − 2a + c + d + e + f − n, −a + e − n, −n; 1) where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T9836 \underline{Description}: Transformation formula ([2], Ex. 3.21(iii), q → 1) in form of a rule. A q-analogue is HYPQ’s T10905.

9F_{8}[ \begin{matrix}a, 1 + \frac{a}{2}, b, c, a − b − c, −A + C − n, −B − C − n, −A + B − n, −n \\ \frac{a}{2}, 1 + a − b, 1 + a − c, 1 + b + c, −C − n, −A + B + C − n, −B − n, −A − n \end{matrix}; 1]

⟶ \frac{(1 + c, 1 + A − B, 1 + A − C, 1 + B + C, 1 + a, 1 + b, 1 + a − b − c)_{n}}{(1 + b + c, 1 + A, 1 + B, 1 + A − B − C, 1 + C, 1 + a − b, 1 + a − c)_{n}}

9F_{8}[ \begin{matrix}A, 1 + \frac{A}{2}, B, C, A − B − C, −a + c − n, −b − c − n, −a + b − n, −n \\ \frac{A}{2}, 1 + A − B, 1 + A − C, 1 + B + C, −c − n, −a + b + c − n, −b − n, −a − n \end{matrix}; 1]

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T9837 \underline{Description}: Transformation formula ([7], (2.4.3.3), reversed) in form of a rule. A q-analogue is HYPQ’s T121104.

9V_{8}(a; c, d, 1 + 2a − b − c − d, 1 + \frac{b}{2}, \frac{1}{2} + \frac{b}{2}, 2a − b + n, −n; 1)

⟶ \frac{(1 + a, 2a − 2b)_{n}}{(a − b, 2a − b)_{n}}_{6}F_{5}[ \begin{matrix}b, 1 + \frac{b}{2}, −a + b + c, −a + b + d, 1 + a − c − d, −n \\ \frac{b}{2}, 1 + a − c, 1 + a − d, −a + b + c + d, 1 − 2a + 2b − n \end{matrix}; 1]

Alphabetic List: T9892 131

T9891 \underline{Description}: Transformation formula ([2], Appendix (III.39), q ↑ 1) in form of a rule. A q-analogue is HYPQ’s T10961.

9V_{8}(a; b, c, d, e, f, g, 2 + 3a − b − c − d − e − f − g; 1)

⟶ Γ [ \begin{matrix}2 + 2a − c − d − e, −1 − 2a + b + c + d + e, 1 + a − f, 1 + a − g, \\ 1 + a, −a + b, 2 + 2a − c − d − e − f, 2 + 2a − c − d − e − g, \end{matrix}

\begin{matrix}−1 − 2a + b + c + d + e + f + g, −a + b + f, −a + b + g, 2 + 2a − c − d − e − f − g \\ −a + b + f + g, −1 − 2a + b + c + d + e + f, −1 − 2a + b + c + d + e + g, 1 + a − f − g \end{matrix} ]

9V_{8}(1 + 2a − c − d − e; b, 1 + a − d − e, 1 + a − c − e, 1 + a − c − d, f, g, 2 + 3a − b − c − d − e − f − g; 1)

− Γ [ \begin{matrix}1 − a + 2b, a − b, 1 + a − c, 1 + a − d, 1 + a − e, 1 + a − f, 1 + a − g, −1 − 2a + b + c + d + e + f + g, \\ 1 + a, −a + b, c, d, e, f, g, 2 + 3a − b − c − d − e − f − g, \end{matrix}

\begin{matrix}−a + b + c, −a + b + d, −a + b + e, −a + b + f, −a + b + g, 2 + 2a − c − d − e − f − g \\ 1 + b − c, 1 + b − d, 1 + b − e, 1 + b − f, 1 + b − g, −1 − 3a + 2b + c + d + e + f + g \end{matrix} ]

9V_{8}(−a + 2b; b, −a + b + c, −a + b + d, −a + b + e, −a + b + f, −a + b + g, 2 + 2a − c − d − e − f − g; 1)

+ Γ [ \begin{matrix}−2a + 2b + c + d + e, 1 + 2a − b − c − d − e, 1 + a − c, 1 + a − d, 1 + a − e, 1 + a − f, \\ 1 + a, −a + b, f, g, 2 + 3a − b − c − d − e − f − g, 1 + b − f, 1 + b − g, \end{matrix}

\begin{matrix}1 + a − g, −1 − 2a + b + c + d + e + f + g, −a + b + c, −a + b + d, −a + b + e, \\ −1 − 3a + 2b + c + d + e + f + g, 1 + a − d − e, 1 + a − c − e, 1 + a − c − d, \end{matrix}

\begin{matrix}−a + b + f, −a + b + g, 2 + 2a − c − d − e − f − g \\ −a + b + d + e, −a + b + c + e, −a + b + c + d \end{matrix} ]_{9}V_{8}(−1 − 2a + 2b + c + d + e; b, −a + b + c, −a + b + d,

− a + b + e, −1 − 2a + b + c + d + e + f, −1 − 2a + b + c + d + e + g, 1 + a − f − g; 1)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T9892 \underline{Description}: Transformation formula ([2], (3.5.7), q → 1) in form of a rule. A q-analogue is HYPQ’s T10962.

9V_{8}(a; b, c, \frac{1}{2} + c, d, \frac{1}{2} + d, e, \frac{1}{2} + e; 1)

⟶ Γ [ \begin{matrix}\frac{1}{2} + a − c, 1 + a − c, \frac{1}{2} + a − d, 1 + a − d, \frac{1}{2} + a − e, 1 + a − e, \frac{1}{2} + a − c − d − e, 1 + a − c − d − e \\ \frac{1}{2} + a, 1 + a, \frac{1}{2} + a − c − d, 1 + a − c − d, \frac{1}{2} + a − c − e, 1 + a − c − e, \frac{1}{2} + a − d − e, 1 + a − d − e \end{matrix} ]

4F_{3}[ \begin{matrix}\frac{1}{2} + a − b, 2c, 2d, 2e \\ \frac{1}{2} + a, 1 + 2a − 2b, −2a + 2c + 2d + 2e \end{matrix}; 1]

+ Γ [ \begin{matrix}1 + a − b, \frac{1}{2} + a − c, 1 + a − c, \frac{1}{2} + a − d, 1 + a − d, \frac{3}{2} + 3a − b − 2c − 2d − 2e, \\ 1 + a, c, \frac{1}{2} + c, d, \frac{1}{2} + d, \frac{3}{2} + 3a − 2c − 2d − 2e, 1 + 2a − b − c − d − e, \end{matrix}

\begin{matrix}\frac{1}{2} + a − e, 1 + a − e, −\frac{1}{2} − a + c + d + e, −a + c + d + e \\ \frac{3}{2} + 2a − b − c − d − e, e, \frac{1}{2} + e \end{matrix} ]

4F_{3}[ \begin{matrix}\frac{3}{2} + 3a − b − 2c − 2d − 2e, 1 + 2a − 2c − 2d, 1 + 2a − 2c − 2e, 1 + 2a − 2d − 2e \\ \frac{3}{2} + 3a − 2c − 2d − 2e, 2 + 4a − 2b − 2c − 2d − 2e, 2 + 2a − 2c − 2d − 2e \end{matrix}; 1]

9V_{8}(a; b, c, d, e, f, g, 2 + 3a − b − c − d − e − f − g; 1)

+ Γ [ \begin{matrix}a − b, 1 − a + 2b, 1 + a − c, −a + b + c, 1 + a − d, −a + b + d, 1 + a − e, −a + b + e, 1 + a − f, \\ 1 + a, −a + b, 1 + b − c, c, 1 + b − d, d, 1 + b − e, e, 1 + b − f, f, 1 + b − g, \end{matrix}

\begin{matrix}−a + b + f, 1 + a − g, 2 + 2a − c − d − e − f − g, −a + b + g, −1 − 2a + b + c + d + e + f + g \\ 2 + 3a − b − c − d − e − f − g, g, −1 − 3a + 2b + c + d + e + f + g \end{matrix} ]

9V_{8}(−a + 2b; b, −a + b + c, −a + b + d, −a + b + e, −a + b + f, −a + b + g, 2 + 2a − c − d − e − f − g; 1)

⟶ Γ [ \begin{matrix}1 + a − c, −a + b + c, 1 + a − d, −a + b + d, 1 + a − e, −a + b + e, 1 + a − f, −a + b + f, \\ 1 + a, −a + b, 1 + b − g, 2 + 2a − c − d − e − g, 2 + 2a − c − d − f − g, 2 + 2a − c − e − f − g, \end{matrix}

\begin{matrix}3 + 3a − c − d − e − f − 2g, 2 + 2a − c − d − e − f − g, \\ 2 + 2a − d − e − f − g, g, −1 − 2a + b + c + d + e + g, −1 − 2a + b + c + d + f + g, \end{matrix}

\begin{matrix}−1 − 2a + b + c + d + e + f + g, −2 − 3a + b + c + d + e + f + 2g \\ −1 − 2a + b + c + e + f + g, −1 − 2a + b + d + e + f + g \end{matrix} ]
9V_{8}(2 + 3a − c − d − e − f − 2g; b, 1 + a − c − g, 1 + a − d − g, 1 + a − e − g, 1 + a − f − g,
2 + 2a − c − d − e − f − g, 2 + 3a − b − c − d − e − f − g; 1)

+ Γ [ \begin{matrix}1 + a − c, −a + b + c, 1 + a − d, −a + b + d, 1 + a − e, −a + b + e, 1 + a − f, −a + b + f, \\ 1 + a, −a + b, 1 + a − c − g, 1 + a − d − g, 1 + a − e − g, 1 + a − f − g, 2 + 3a − b − c − d − e − f − g, \end{matrix}

\begin{matrix}2 + 3a − b − c − d − e − f − 2g, 1 + a − g, −a + b + g, −1 − 3a + 2b + c + d + e + f + 2g \\ −a + b + c + g, −a + b + d + g, −a + b + e + g, −a + b + f + g, −1 − 3a + 2b + c + d + e + f + g \end{matrix} ]

9V_{8}(−2 − 3a + 2b + c + d + e + f + 2g; b, −1 − 2a + b + d + e + f + g, −1 − 2a + b + c + e + f + g, − 1 − 2a + b + c + d + f + g, −1 − 2a + b + c + d + e + g, −a + b + g, g; 1)

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T9894 \underline{Description}: Transformation formula ([2], Ex. 2.25, q → 1, reversed) in form of a rule. A q-analogue is HYPQ’s

Alphabetic List: T111032 133

  T121161.

9V_{8}(a; b, \frac{1}{2} + b, c, d, 1 + 2a − 2b − c − d, e, 1 + 2a − 2b − e; 1)

+ Γ [ \begin{matrix}1 + a − 2b, 1 + 2a − 2b, 1 + a − c, 1 + a − d, −a + 2b + c + d, 3 + 3a − 4b − 2e, 1 + a − e, 1 + a − 2b + c − e, \\ 1 + a, 2b, c, 1 + 2a − 2b − c − d, d, 3 + 4a − 6b − 2e, 1 + a − 2b − e, \end{matrix}

\begin{matrix}2 + 3a − 4b − c − d − e, 1 + a − 2b + d − e, −1 − a + 2b + e \\ 2 + 2a − 2b − c − e, 2 + 2a − 2b − d − e, 1 + c + d − e, e \end{matrix} ]

9V_{8}(2 + 3a − 4b − 2e; 1 + a − b − e, \frac{3}{2} + a − b − e, 1 + a − 2b, 1 + 2a − 2b − e, 1 + a − 2b + c − e, 1 + a − 2b + d − e,

2 + 3a − 4b − c − d − e; 1)

⟶ Γ [ \begin{matrix}1 + a − 2b, 1 + 2a − 2b, 1 + a − e, 1 + 2a − 4b − e \\ 1 + a, 1 + 2a − 4b, 1 + a − 2b − e, 1 + 2a − 2b − e \end{matrix} ]

5F_{4}[ \begin{matrix}2b, −a + 2b + c, −a + 2b + d, 1 + a − c − d, e \\ 1 + a − c, 1 + a − d, −a + 2b + c + d, −2a + 4b + e \end{matrix}; 1]

+ Γ [ \begin{matrix}1 + a − 2b, 1 + 2a − 2b, 1 + a − c, 1 + a − d, −a + 2b + c + d, 1 + a − e, 1 + a − 2b + c − e, \\ 1 + a, 2b, −a + 2b + c, 1 + a − c − d, −a + 2b + d, 1 + a − 2b − e, \end{matrix}

\begin{matrix}2 + 3a − 4b − c − d − e, 1 + a − 2b + d − e, −1 − 2a + 4b + e \\ 2 + 3a − 4b − c − e, 2 + 3a − 4b − d − e, 1 + a − 2b + c + d − e, e \end{matrix} ]

5F_{4}[ \begin{matrix}1 + 2a − 4b, 1 + 2a − 2b − e, 1 + a − 2b + c − e, 1 + a − 2b + d − e, 2 + 3a − 4b − c − d − e \\ 2 + 2a − 4b − e, 2 + 3a − 4b − c − e, 2 + 3a − 4b − d − e, 1 + a − 2b + c + d − e \end{matrix}; 1]

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T111031 \underline{Description}: Transformation formula ([6], (7.8), q → 1) in form of a rule. A q-analogue is HYPQ’s T121105.

11V_{10}(a; c, \frac{1}{2} + c, d, \frac{1}{2} + d, e, \frac{1}{2} + e, \frac{1}{2} + 2a − c − d − e + \frac{n}{2}, 1 + 2a − c − d − e + \frac{n}{2}, −n; 1)

⟶ \frac{(1 + 2a, 1 + 2a − 2c − 2d, 1 + 2a − 2c − 2e, 1 + 2a − 2d − 2e)_{n}}{(1 + 2a − 2c, 1 + 2a − 2d, 1 + 2a − 2e, 1 + 2a − 2c − 2d − 2e)_{n}}

8V_{7}(2a + n; 2c, 2d, 2e, \frac{1}{2} + a + n, 1 + 4a − 2c − 2d − 2e + n, −n; −1)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

T111032 \underline{Description}: Transformation formula ([6], (7.7), q → 1) in form of a rule. A q-analogue is HYPQ’s T121106.

11V_{10}(a; 2e, c, \frac{1}{2} + c, d, \frac{1}{2} + d, \frac{1}{2} + 2a − c − d − e + \frac{n}{2}, 1 + 2a − c − d − e + \frac{n}{2}, \frac{1}{2} − \frac{n}{2}, \frac{−n}{2}; 1)

⟶ \frac{(1 + 2a, 1 + 2a − 2c − 2d, 1 + 2a − 2c − 2e, 1 + 2a − 2d − 2e)_{n}}{(1 + 2a − 2c, 1 + 2a − 2d, 1 + 2a − 2e, 1 + 2a − 2c − 2d − 2e)_{n}}

8V_{7}(2a − 2e; \frac{1}{2} + a − 2e, 2c, 2d, 2e, 1 + 4a − 2c − 2d − 2e + n, −n; −1)

where n is a nonnegative integer.

\underline{See also}: S2103, S3201, TListe, TransListe, Ers, PosListe.

134 Alphabetic List: TeXFV

TeX \underline{Description}: Switch that changes the output of TeXForm to be usable with Plain-TEX and LaTEX. By default the

output of TeXForm is usable with AMS-TEX. \underline{Usage}: TeX. \underline{Example(s)}:

In[1]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
In[2]:= TeXForm[F[{a,b},{c},z]]
Out[2]//TeXForm=
  {} {2} F {1} \!\left [ \matrix { a, b}\\ { c}\endmatrix ; {\displaystyle
   z}\right ]
In[3]:= TeX
In[4]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with Plain-TeX and LaTeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
In[5]:= TeXForm[F[{a,b},{c},z]]
Out[5]//TeXForm=
  {} {2} F {1} \!\left [ \matrix { a, b\cr c} ; {\displaystyle z} \right ]
\underline{See also}: AmSTeX, AmSLaTeX, LaTeX, TeXMat, TeXFV.

TeXFV \underline{Description}: Switch that toggles between writing very well-poised hypergeometric series in terms of V and in terms of

F, respectively, when written in TeXForm. By default very well-poised hypergeometric series are written in terms of V.

\underline{Usage}: TeXFV.

\underline{Example(s)}:

In[1]:= hypAttributes

Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
In[2]:= F[{a,1+a/2,b,c},{a/2,a+1-b,a+1-b},z]
                        a
                 a, 1 + -, b, c
                        2
Out[2]=  F                          ; z
        4 3  a
             -, 1 + a - b, 1 + a - b
             2
In[3]:= Reset
Out[0]= 0
In[1]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
In[2]:= F[{a,1+a/2,b,c},{a/2,a+1-b,a+1-c},z]
                        a
                 a, 1 + -, b, c
                        2
Out[2]=  F                          ; z
        4 3  a
             -, 1 + a - b, 1 + a - c
             2
In[3]:= TeXForm[%]
Out[3]//TeXForm= {} {4} V {3} ({\displaystyle a; b, c}; {\displaystyle z})
In[4]:= TeXFV
In[5]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses F[] for very well-poised basic hypergeometric series.

In[6]:= TeXForm[%2]
Out[6]//TeXForm=
  {} {4} F {3} \!\left [ \matrix { a, 1 + {a\over 2}, b, c}\\ { {a\over 2},
   1 + a - b, 1 + a - c}\endmatrix ; {\displaystyle z}\right ]
In[7]:= TeXFV
In[8]:= hypAttributes
Automatic evaluation of p and F is inactive.
Automatic cancelling in F is active.
The output of TeXForm can be used with AmS-TeX.
TeXForm uses V[] for very well-poised basic hypergeometric series.
\underline{See also}: F, V, hypAttributes.

TeXMat

\underline{Description}: Function that writes (to be precise: appends) an expression Expr in InputForm to a file [name].m and the TeXForm of Expr to the file [name].tex. The expressions are numbered automatically. The number can be reset by SchreibeZahl. The string comment is optional. It allows to place the comment comment above the expression and the number in each of the two files.

\underline{Usage}: TeXMat[Expr,name,comment].

\underline{Example(s)}:

In[1]:= TeXMat[p[a/3,2*n],filename]
In[2]:= TeXMat[F[{a,b},{c},z],filename,"A hypergeometric  2F 1 series"]
In[3]:= !type filename.m
A[1]:=
p[a/3, 2*n]
"A hypergeometric  2F 1 series"
A[2]:=
F[{a, b}, {c}, z]
In[3]:= !type filename.tex
A[1]:=
({ \textstyle {a\over 3}}) {2\,n}
"A hypergeometric  2F 1 series"
A[2]:=
{} {2} F {1} \!\left [ \matrix { a, b}\\ { c}\endmatrix ; {\displaystyle
  z}\right ]
\underline{See also}: AmSTeX, AmSLaTeX, LaTeX, TeX, TeXFV, SchreibeZahl.

Alphabetic List: Tgl2132 137

Tgl2103 \underline{Description}: Transformation formula ([7], (1.3.15)) in form of an equation. It is the same transformation as that in

  T2103.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2104 \underline{Description}: Transformation formula ([2], Appendix (III.4), q ↑ 1) in form of an equation. It is the same transformation

as that in T2104.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2106 \underline{Description}: Transformation formula ([2], Appendix (III.6), q ↑ 1) in form of an equation. It is the same transformation

as that in T2106.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2107 \underline{Description}: Transformation formula ([2], Appendix (III.7), q ↑ 1) in form of an equation. It is the same transformation

as that in T2107.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2110 \underline{Description}: Transformation formula ([6], (3.2)) in form of an equation. It is the same transformation as that in

  T2110.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2112 \underline{Description}: Transformation formula ([6], (5.10)) in form of an equation. It is the same transformation as that in

  T2112.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2131 \underline{Description}: Transformation formula ([2], Appendix (III.6), reversed, q ↑ 1) in form of an equation. It is the same

transformation as that in T2131.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2132 \underline{Description}: Transformation formula ([7], (2.5.7)) in form of an equation. It is the same transformation as that in

  T2132.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

138 Alphabetic List: Tgl2140

Tgl2133 \underline{Description}: Transformation formula ([6], (5.12)) in form of an equation. It is the same transformation as that in

  T2133.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2134 \underline{Description}: Transformation formula ([1], Ex. 4.(iii), p. 97, reversed) in form of an equation. It is the same

transformation as that in T2134.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2135 \underline{Description}: Transformation formula ([1], Ex. 4.(iii), p. 97) in form of an equation. It is the same transformation as

that in T2135.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2136 \underline{Description}: Transformation formula ([6], (5.10), reversed) in form of an equation. It is the same transformation as

that in T2136.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2137 \underline{Description}: Transformation formula ([6], (5.12), reversed) in form of an equation. It is the same transformation as

that in T2137.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2138 \underline{Description}: Transformation formula ([6], (3.31), reversed) in form of an equation. It is the same transformation as

that in T2138.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2139 \underline{Description}: Transformation formula ([6], (3.31)) in form of an equation. It is the same transformation as that in

  T2139.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2140 \underline{Description}: Transformation formula ([6], (3.2), reversed) in form of an equation. It is the same transformation as

that in T2140.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tgl3207 139

Tgl2141 \underline{Description}: Transformation formula ([2], (3.4.8), q → 1, reversed) in form of an equation. It is the same transformation

as that in T2141.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2163 \underline{Description}: Transformation formula ([2], Ex 3.8, q → 1; [7], pp. 36/37) in form of an equation. It is the same

transformation as that in T2163.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2191 \underline{Description}: Transformation formula ([7], (1.8.10)) in form of an equation. It is the same transformation as that in

  T2191.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl2192 \underline{Description}: Transformation formula ([2], Ex 3.8, q → 1, reversed; [7], pp. 36/37) in form of an equation. It is the

same transformation as that in T2192.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3204 \underline{Description}: Transformation formula ([1], Ex. 7, p. 98) in form of an equation. It is the same transformation as that

  in T3204.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3205 \underline{Description}: Transformation formula ([7], (2.3.3.7)) in form of an equation. It is the same transformation as that in

  T3205.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3206 \underline{Description}: Transformation formula ([1], Ex. 7, p. 98, terminating form) in form of an equation. It is the same

transformation as that in T3206.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3207 \underline{Description}: Transformation formula (Thomae [1879] in [2] (3.1.1)) in form of an equation. It is the same transforma-

tion as that in T3207.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

140 Alphabetic List: Tgl3237

Tgl3217 \underline{Description}: Transformation formula ([1] 4.4(2), reversed) in form of an equation. It is the same transformation as

that in T3217.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3231 \underline{Description}: Transformation formula ([2], Appendix (III.21), q ↑ 1) in form of an equation. It is the same transforma-

tion as that in T3231.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3232 \underline{Description}: Transformation formula ([1], Ex. 4.(iv), p.97, reversed, first form) in form of an equation. It is the same

transformation as that in T3232.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3233 \underline{Description}: Transformation formula ([1], Ex. 4.(iv), p.97, reversed, second form) in form of an equation. It is the

same transformation as that in T3233.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3234 \underline{Description}: Transformation formula ([7], (2.5.7), reversed) in form of an equation. It is the same transformation as

that in T3234.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3235 \underline{Description}: Transformation formula ([2], Ex 3.4, q → 1, reversed) in form of an equation. It is the same transformation

as that in T3235.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3236 \underline{Description}: Transformation formula ([2], (3.4.8), q → 1) in form of an equation. It is the same transformation as

that in T3236.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3237 \underline{Description}: Transformation formula ([2], (3.10.4), q → 1, reversed) in form of an equation. It is the same transfor-

mation as that in T3237.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tgl3267 141

Tgl3238 \underline{Description}: Transformation formula ([1], Ex. 6, p. 97) in form of an equation. It is the same transformation as that

  in T3238.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3239 \underline{Description}: Transformation formula ([1], Ex. 4.(iv), p. 97) in form of an equation. It is the same transformation as

that in T3239.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3240 \underline{Description}: Transformation formula ([2], (3.5.10), q → 1, reversed) in form of an equation. It is the same transfor-

mation as that in T3240.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3261 \underline{Description}: Transformation formula ([2], (III.33), q ↑ 1) in form of an equation. It is the same transformation as that

  in T3261.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3262 \underline{Description}: Transformation formula ([7], (4.3.4.2)) in form of an equation. It is the same transformation as that in

  T3262.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3263 \underline{Description}: Transformation formula ([2], Appendix (III.33), q → 1, reversed) in form of an equation. It is the same

transformation as that in T3263.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3264 \underline{Description}: Transformation formula ([2], Appendix (III.34), q → 1, reversed) in form of an equation. It is the same

transformation as that in T3264.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl3267 \underline{Description}: Transformation formula ([2], Ex. 3.6, q → 1) in form of an equation. It is the same transformation as

that in T3267.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

142 Alphabetic List: Tgl4310

Tgl3268 \underline{Description}: Transformation formula ([2], Ex. 3.6, q → 1, reversed) in form of an equation. It is the same transfor-

mation as that in T3268.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4301 \underline{Description}: Transformation formula ([7], (4.3.5.1)) in form of an equation. It is the same transformation as that in

  T4301.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4302 \underline{Description}: Transformation formula ([2], Appendix (III.16), q ↑ 1) in form of an equation. It is the same transforma-

tion as that in T4302.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4303 \underline{Description}: Transformation formula ([7], (2.4.1.1)) in form of an equation. It is the same transformation as that in

  T4303.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4304 \underline{Description}: Transformation formula ([7], (4.3.6.4)) in form of an equation. It is the same transformation as that in

  T4304.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4306 \underline{Description}: Transformation formula ([2], Appendix (III.21), q ↑ 1, reversed) in form of an equation. It is the same

transformation as that in T3231.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4309 \underline{Description}: Transformation formula ([2], Ex. 2.13(i), q → 1) in form of an equation. It is the same transformation

as that in T4309.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4310 \underline{Description}: Transformation formula ([2], Ex. 2.13(ii), q → 1) in form of an equation. It is the same transformation

as that in T4310.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tgl5403 143

Tgl4312 \underline{Description}: Transformation formula ([2], Ex. 3.4, q → 1) in form of an equation. It is the same transformation as

that in T4312.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4331 \underline{Description}: Transformation formula ([1], Ex. 6, p. 97, reversed) in form of an equation. It is the same transformation

as that in T4331.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4332 \underline{Description}: Transformation formula ([1], 4.6(1)) in form of an equation. It is the same transformation as that in

  T4332.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4362 \underline{Description}: Transformation formula ([7], (2.4.4.3), reversed) in form of an equation. It is the same transformation as

that in T4362.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl4391 \underline{Description}: Transformation formula ([2], (3.5.7), q → 1, reversed) in form of an equation. It is the same transformation

as that in T4391.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5401 \underline{Description}: Transformation formula ([7], (2.4.3.4)) in form of an equation. It is the same transformation as that in

  T5401.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5402 \underline{Description}: Transformation formula ([2], Appendix (III.26), q ↑ 1) in form of an equation. It is the same transforma-

tion as that in T5402.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl5403 \underline{Description}: Transformation formula ([1], 4.6(1), reversed) in form of an equation. It is the same transformation as

that in T5403.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

144 Alphabetic List: Tgl7632

Tgl5468 \underline{Description}: Transformation formula ([2], Ex. 2.25, q → 1) in form of an equation. It is the same transformation as

that in T5468.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl6501 \underline{Description}: Transformation formula ([7], (2.4.3.3)) in form of an equation. It is the same transformation as that in

  T6501.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl6531 \underline{Description}: Transformation formula ([7], (2.4.3.5)) in form of an equation. It is the same transformation as that in

  T6531.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl6532 \underline{Description}: Transformation formula ([2], Ex. 2.13(ii), q → 1, reversed) in form of an equation. It is the same

transformation as that in T6532.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl6533 \underline{Description}: Transformation formula ([2], (3.10.4), q → 1) in form of an equation. It is the same transformation as

that in T6533.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl6534 \underline{Description}: Transformation formula ([1], 4.4(2)) in form of an equation. It is the same transformation as that in

  T6534.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl7631 \underline{Description}: Transformation formula ([7], (2.4.1.1), reversed) in form of an equation. It is the same transformation as

that in T7631.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl7632 \underline{Description}: Transformation formula ([7], (2.4.1.1), reversed) in form of an equation. It is the same transformation as

that in T7632.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tgl7693 145

Tgl7633 \underline{Description}: Transformation formula ([7], (4.3.6.4), reversed) in form of an equation. It is the same transformation as

that in T7633.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl7634 \underline{Description}: Transformation formula ([1], 7.5.(1)) in form of an equation. It is the same transformation as that in

  T7634.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl7635 \underline{Description}: Transformation formula ([1], 7.5.(2)) in form of an equation. It is the same transformation as that in

  T7635.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl7636 \underline{Description}: Transformation formula ([2], (3.5.10), q → 1) in form of an equation. It is the same transformation as

that in T7636.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl7637 \underline{Description}: Transformation formula ([2], Ex. 2.13(i), q → 1, reversed) in form of an equation. It is the same

transformation as that in T7637.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl7691 \underline{Description}: Transformation formula ([7], (2.4.4.3)) in form of an equation. It is the same transformation as that in

  T7691.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl7692 \underline{Description}: Transformation formula ([7], (4.3.7.8)) in form of an equation. It is the same transformation as that in

  T7692.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl7693 \underline{Description}: Transformation formula ([2], Ex. 2.15, q → 1) in form of an equation. It is the same transformation as

that in T7693.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

146 Alphabetic List: Tgl9835

Tgl7694 \underline{Description}: Transformation formula ([7], (4.3.7.8), reversed) in form of an equation. It is the same transformation as

that in T7694.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8731 \underline{Description}: Transformation formula ([6], (7.7), q → 1, reversed) in form of an equation. It is the same transformation

as that in T8731.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl8732 \underline{Description}: Transformation formula ([6], (7.8), q → 1, reversed) in form of an equation. It is the same transformation

as that in T8732.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl9831 \underline{Description}: Transformation formula ([7], (2.4.4.1)) in form of an equation. It is the same transformation as that in

  T9831.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl9832 \underline{Description}: Transformation formula ([7], (2.4.3.4), reversed) in form of an equation. It is the same transformation as

that in T9832.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl9833 \underline{Description}: Transformation formula ([2], Appendix (III.26), q ↑ 1, reversed) in form of an equation. It is the same

transformation as that in T9833.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl9834 \underline{Description}: Transformation formula ([7], (2.4.3.5), reversed) in form of an equation. It is the same transformation as

that in T9834.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl9835 \underline{Description}: Transformation formula ([1], 7.6(1)) in form of an equation. It is the same transformation as that in

  T9835.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Alphabetic List: Tgl111032 147

Tgl9836 \underline{Description}: Transformation formula ([2], Ex. 3.21(iii), q → 1) in form of an equation. It is the same transformation

as that in T9836.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl9837 \underline{Description}: Transformation formula ([7], (2.4.3.3), reversed) in form of an equation. It is the same transformation as

that in T9837.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl9891 \underline{Description}: Transformation formula ([2], Appendix (III.39), q ↑ 1) in form of an equation. It is the same transforma-

tion as that in T9891.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl9892 \underline{Description}: Transformation formula ([2], (3.5.7), q → 1) in form of an equation. It is the same transformation as

that in T9892.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl9893 \underline{Description}: Transformation formula ([2], Ex. 2.30, q → 1) in form of an equation. It is the same transformation as

that in T9893.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl9894 \underline{Description}: Transformation formula ([2], Ex. 2.25, q → 1, reversed) in form of an equation. It is the same transfor-

mation as that in T9894.

\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl111031 \underline{Description}: Transformation formula ([6], (7.8), q → 1) in form of an equation. It is the same transformation as that

  in T111031.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tgl111032 \underline{Description}: Transformation formula ([6], (7.7), q → 1) in form of an equation. It is the same transformation as that

  in T111032.
\underline{See also}: Sgl2101, TransListe$gl, Gleichung.

Tli2103

\underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7], (1.3.15)) in form of a rule. The transformation is the same as that in T2103.

\underline{See also}: Tli2107, Tli2104, TransListe$gl, Gleichung.

Tli2104

\underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Appendix (III.4), q ↑ 1) in form of a rule. The transformation is the same as that in T2104.

\underline{Example(s)}:

Here we demonstrate the iterated application of Tli-rules. We continue the Tli2107-example. For information of how to read the resulting listing confer Tli2107.

In[4]:= %2/.Tli2104

                 -m, -n
Out[4]= {2,  F         ; z  , {{{{
            2 1    a

           m      -m, 1 - a - m    1 - z
          z   F                ; -(-----)   (a + n)
             2 1  1 - a - m - n      z             m
⟩  ------------------------------------------}}, T2107}, T2104},
                             (a)
                                m

          n      -n, 1 - a - n    1 - z
         z   F                ; -(-----)   (a + n)
            2 1  1 - a - m - n      z             m
⟩  {{{{------------------------------------------}}, T2107},
                            (a)
                               m
⟩  {FPerm[2, 1, u], T2104}},

         m      -m, 1 - a - m    1 - z
        z   F                ; -(-----)   (a + m)
           2 1  1 - a - m - n      z             n
⟩  {{{------------------------------------------},
                           (a)
                              n

⟩  {FPerm[2, 1, u], T2107}}, {FPerm[2, 1, u], T2104}},

         n      -n, 1 - a - n    1 - z
        z   F                ; -(-----)   (a + m)
           2 1  1 - a - m - n      z             n
⟩  {{{------------------------------------------},
                           (a)
                              n
⟩  {FPerm[2, 1, u], T2107}}, T2104}}
In[5]:= %1/.FPerm[2,1,u]/.T2107/.FPerm[2,1,u]/.T2104
Is n a nonnegative integer?
[y|n]:    y

         m      -m, 1 - a - m    1 - z
        z   F                ; -(-----)   (a + m)
           2 1  1 - a - m - n      z             n
Out[5]= ------------------------------------------
                           (a)
                              n
\underline{See also}: Tli2107, TransListe, Gleichung.

Tli2106

\underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Appendix (III.6), q ↑ 1) in form of a rule. The transformation is the same as that in T2106.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2107

\underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Appendix (III.7), q ↑ 1) in form of a rule. The transformation is the same as that in T2107.

\underline{Example(s)}:

In[1]:= F[{-m,-n},{a},z]

             -m, -n
Out[1]=  F         ; z
        2 1    a

In[2]:= %/.Tli2107
Is m a nonnegative integer?
[y|n]:    y
Is n a nonnegative integer?
[y|n]:    y

                                         -m, -n
                                  F                ; 1 - z   (a + n)
                                 2 1  1 - a - m - n                 m
                 -m, -n
Out[2]= {1,  F         ; z  , {{{------------------------------------}},
            2 1    a                             (a)
                                                    m

                        -n, -m
                 F                ; 1 - z   (a + m)
                2 1  1 - a - m - n                 n
⟩  T2107}, {{------------------------------------},
                                (a)
                                   n
⟩  {FPerm[2, 1, u], T2107}}}

The first entry in this list counts the number of iterations of Tli-rules (cf. Tli2104), the second entry displays the original expression to which the Tli-rules are applied. The subsequent entries of the list always display an expression together with the sequence of rules that have to be applied to obtain this expression from the original expression. For

instance, the above list says that the number of iterations is 1, the original expression is _{2}F_{1}[ \begin{matrix}−m, −n \\ a \end{matrix}; z], and (if the

parameters of the original series are permuted) by the application of T2101 two different expressions can be obtained from the original series. The second of them is obtained by first permuting the upper parameters by FPerm[2,1,u] and then applying T2101.

In[3]:= %1/.FPerm[2,1,u]/.T2107
Is n a nonnegative integer?
[y|n]:    y

                -n, -m
         F                ; 1 - z   (a + m)
        2 1  1 - a - m - n                 n
Out[3]= ------------------------------------
                        (a)
                           n

For an example of how to iterate Tli-rules see Tli2104.

\underline{See also}: Tli2104, TransListe, Gleichung.

Alphabetic List: Tli2136 151

Tli2110 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (3.2))

in form of a rule. The transformation is the same as that in T2110.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2112 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (5.10))

in form of a rule. The transformation is the same as that in T2112.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2131 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

Appendix (III.6), reversed, q ↑ 1) in form of a rule. The transformation is the same as that in T2131.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2132 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7], (2.5.7))

in form of a rule. The transformation is the same as that in T2132.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2133 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (5.12))

in form of a rule. The transformation is the same as that in T2133.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2134 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex.

4.(iii), p. 97, reversed) in form of a rule. The transformation is the same as that in T2134.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2135 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex.

4.(iii), p. 97) in form of a rule. The transformation is the same as that in T2135.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2136 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (5.10),

reversed) in form of a rule. The transformation is the same as that in T2136.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

152 Alphabetic List: Tli2192

Tli2137 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (5.12),

reversed) in form of a rule. The transformation is the same as that in T2137.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2138 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (3.31),

reversed) in form of a rule. The transformation is the same as that in T2138.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2139 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (3.31))

in form of a rule. The transformation is the same as that in T2139.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2140 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (3.2),

reversed) in form of a rule. The transformation is the same as that in T2140.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2141 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], (3.4.8),

q → 1, reversed) in form of a rule. The transformation is the same as that in T2141.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2163 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex

3.8, q → 1; [7], pp. 36/37) in form of a rule. The transformation is the same as that in T2163.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2191 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(1.8.10)) in form of a rule. The transformation is the same as that in T2191.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli2192 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex

3.8, q → 1, reversed; [7], pp. 36/37) in form of a rule. The transformation is the same as that in T2192.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli3233 153

Tli3204 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 7,

p. 98) in form of a rule. The transformation is the same as that in T3204.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3205 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.3.3.7)) in form of a rule. The transformation is the same as that in T3205.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3206 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex. 7,

p. 98, terminating form) in form of a rule. The transformation is the same as that in T3206.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3207 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula (Thomae

[1879] in [2] (3.1.1)) in form of a rule. The transformation is the same as that in T3207.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3217 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1] 4.4(2),

reversed) in form of a rule. The transformation is the same as that in T3217.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3231 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

Appendix (III.21), q ↑ 1) in form of a rule. The transformation is the same as that in T3231.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3232 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 4.(iv), p.97, reversed, first form) in form of a rule. The transformation is the same as that in T3232.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3233 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 4.(iv), p.97, reversed, second form) in form of a rule. The transformation is the same as that in T3233.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

154 Alphabetic List: Tli3261

Tli3234 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7], (2.5.7),

reversed) in form of a rule. The transformation is the same as that in T3234.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3235 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex

3.4, q → 1, reversed) in form of a rule. The transformation is the same as that in T3235.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3236 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], (3.4.8),

q → 1) in form of a rule. The transformation is the same as that in T3236.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3237 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

(3.10.4), q → 1, reversed) in form of a rule. The transformation is the same as that in T3237.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3238 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex.

6, p. 97) in form of a rule. The transformation is the same as that in T3238.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3239 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1],

Ex. 4.(iv), p. 97) in form of a rule. The transformation is the same as that in T3239.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3240 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

(3.5.10), q → 1, reversed) in form of a rule. The transformation is the same as that in T3240.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3261 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

(III.33), q ↑ 1) in form of a rule. The transformation is the same as that in T3261.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli4303 155

Tli3262 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(4.3.4.2)) in form of a rule. The transformation is the same as that in T3262.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3263 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

Appendix (III.33), q → 1, reversed) in form of a rule. The transformation is the same as that in T3263.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3264 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

Appendix (III.34), q → 1, reversed) in form of a rule. The transformation is the same as that in T3264.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3267 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

3.6, q → 1) in form of a rule. The transformation is the same as that in T3267.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli3268 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

3.6, q → 1, reversed) in form of a rule. The transformation is the same as that in T3268.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli4301 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(4.3.5.1)) in form of a rule. The transformation is the same as that in T4301.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli4302 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

Appendix (III.16), q ↑ 1) in form of a rule. The transformation is the same as that in T4302.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli4303 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.1.1)) in form of a rule. The transformation is the same as that in T4303.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

156 Alphabetic List: Tli4362

Tli4304 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(4.3.6.4)) in form of a rule. The transformation is the same as that in T4304.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli4306 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

Appendix (III.21), q ↑ 1, reversed) in form of a rule. The transformation is the same as that in T3231.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli4309 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

2.13(i), q → 1) in form of a rule. The transformation is the same as that in T4309.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli4310 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

2.13(ii), q → 1) in form of a rule. The transformation is the same as that in T4310.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli4312 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

3.4, q → 1) in form of a rule. The transformation is the same as that in T4312.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli4331 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], Ex.

6, p. 97, reversed) in form of a rule. The transformation is the same as that in T4331.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli4332 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], 4.6(1))

in form of a rule. The transformation is the same as that in T4332.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli4362 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.4.3), reversed) in form of a rule. The transformation is the same as that in T4362.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli6532 157

Tli4391 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], (3.5.7),

q → 1, reversed) in form of a rule. The transformation is the same as that in T4391.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli5401 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.3.4)) in form of a rule. The transformation is the same as that in T5401.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli5402 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

Appendix (III.26), q ↑ 1) in form of a rule. The transformation is the same as that in T5402.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli5403 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], 4.6(1),

reversed) in form of a rule. The transformation is the same as that in T5403.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli5468 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

2.25, q → 1) in form of a rule. The transformation is the same as that in T5468.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli6501 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.3.3)) in form of a rule. The transformation is the same as that in T6501.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli6531 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.3.5)) in form of a rule. The transformation is the same as that in T6531.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli6532 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

2.13(ii), q → 1, reversed) in form of a rule. The transformation is the same as that in T6532.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

158 Alphabetic List: Tli7636

Tli6533 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

(3.10.4), q → 1) in form of a rule. The transformation is the same as that in T6533.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli6534 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], 4.4(2))

in form of a rule. The transformation is the same as that in T6534.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli7631 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.1.1), reversed) in form of a rule. The transformation is the same as that in T7631.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli7632 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.1.1), reversed) in form of a rule. The transformation is the same as that in T7632.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli7633 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(4.3.6.4), reversed) in form of a rule. The transformation is the same as that in T7633.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli7634 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], 7.5.(1))

in form of a rule. The transformation is the same as that in T7634.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli7635 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], 7.5.(2))

in form of a rule. The transformation is the same as that in T7635.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli7636 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

(3.5.10), q → 1) in form of a rule. The transformation is the same as that in T7636.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Alphabetic List: Tli9831 159

Tli7637 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

2.13(i), q → 1, reversed) in form of a rule. The transformation is the same as that in T7637.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli7691 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.4.3)) in form of a rule. The transformation is the same as that in T7691.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli7692 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(4.3.7.8)) in form of a rule. The transformation is the same as that in T7692.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli7693 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

2.15, q → 1) in form of a rule. The transformation is the same as that in T7693.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli7694 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(4.3.7.8), reversed) in form of a rule. The transformation is the same as that in T7694.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli8731 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (7.7),

q → 1, reversed) in form of a rule. The transformation is the same as that in T8731.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli8732 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (7.8),

q → 1, reversed) in form of a rule. The transformation is the same as that in T8732.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli9831 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.4.1)) in form of a rule. The transformation is the same as that in T9831.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

160 Alphabetic List: Tli9892

Tli9832 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.3.4), reversed) in form of a rule. The transformation is the same as that in T9832.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli9833 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

Appendix (III.26), q ↑ 1, reversed) in form of a rule. The transformation is the same as that in T9833.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli9834 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.3.5), reversed) in form of a rule. The transformation is the same as that in T9834.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli9835 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([1], 7.6(1))

in form of a rule. The transformation is the same as that in T9835.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli9836 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

3.21(iii), q → 1) in form of a rule. The transformation is the same as that in T9836.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli9837 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([7],

(2.4.3.3), reversed) in form of a rule. The transformation is the same as that in T9837.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli9891 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2],

Appendix (III.39), q ↑ 1) in form of a rule. The transformation is the same as that in T9891.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli9892 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], (3.5.7),

q → 1) in form of a rule. The transformation is the same as that in T9892.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Alphabetic List: TListe 161

Tli9893 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

2.30, q → 1) in form of a rule. The transformation is the same as that in T9893.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli9894 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([2], Ex.

2.25, q → 1, reversed) in form of a rule. The transformation is the same as that in T9894.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli111031 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (7.8),

q → 1) in form of a rule. The transformation is the same as that in T111031.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

Tli111032 \underline{Description}: Rule that gives a list of all possible outcomes under application of the transformation formula ([6], (7.7),

q → 1) in form of a rule. The transformation is the same as that in T111032.

\underline{See also}: Tli2107, Tli2104, TransListe, Gleichung.

TListe \underline{Description}: Rule that gives for a hypergeometric series a list of applicable transformation formulas. Each entry

  of this list has the format {ArgumentPermutations,T⟨number⟩}, where ArgumentPermutations is a sequence of

reorderings of the parameters of the hypergeometric series (given in terms of FPerm and FTausche) and T⟨number⟩ is the name of the transformation in form of a rule which can be applied subsequently. You should be aware that TListe automatically applies FOrdne before checking which transformation could be applied.

\underline{Important Note}: If the value returned by TListe is the empty set this does not mean that no transformation can be applied. You always must remember that the list of transformations included in this package is a list of basic transformations. There are numerous special cases of these transformations which are not contained in this list as a separate transformation. The examples below should illustrate these remarks.

\underline{Usage}: Expr/.TListe.

\underline{Example(s)}:

In[1]:= F[{a,b},{c},z]
             a, b
Out[1]=  F       ; z
        2 1   c
In[2]:= %/.TListe
Is -a a nonnegative integer?
[y|n]:    n
Is -b a nonnegative integer?
[y|n]:    n

Is -a a nonnegative integer?
[y|n]:    n
Is -b a nonnegative integer?
[y|n]:    n
Is -b a nonnegative integer?
[y|n]:    n
Is -a a nonnegative integer?
[y|n]:    n
Be sure to apply "FOrdne" before using the following information!
Out[2]= {{T2103}, {T2104}, {T2191}}
In[3]:= F[{-n,2*a,2*b},{-2*n,1/2+a+b},1]
              -n, 2 a, 2 b
Out[3]=  F         1        ; 1
        3 2  -2 n, - + a + b
                   2
In[4]:= %/.TListe
Is -2*b a nonnegative integer?
[y|n]:    n
Is -2*a a nonnegative integer?
[y|n]:    n
Is n a nonnegative integer?
[y|n]:    y
Is -2*b a nonnegative integer?
[y|n]:    n
Is -2*a a nonnegative integer?
[y|n]:    n
Is n a nonnegative integer?
[y|n]:    y
Is -2*a a nonnegative integer?
[y|n]:    n
Is -2*b a nonnegative integer?
[y|n]:    n
Is n a nonnegative integer?
[y|n]:    y
Be sure to apply "FOrdne" before using the following information!
Out[4]= {{T3204}, {T3205}, {T3206}, {T3207}, {T3217}, {T3261}, {T3262},
⟩  {FPerm[2,3,1,u], FTausche[1,2,l], T3231}}
In[5]:= %3/.FOrdne/.FPerm[2,3,1,u]/.FTausche[1,2,l]/.T3231
Is -2*a a nonnegative integer?
[y|n]:    n

Is -2*b a nonnegative integer?
[y|n]:    n
Is n a nonnegative integer?
[y|n]:    y
                  a, b, -n, -n
Out[5]=  F   1              1 - 2 n; 1
        4 3  - + a + b, -n, -------
             2                 2

Now we consider two examples illustrating the note above. Though none of the implemented transformations can be applied, both series can be transformed, the first by a limiting case of Whipple’s transformation, the second by a number of specialized very well-poised _{7}ϕ_{6} transformations. These facts are also observed by using this package.

In[6]:= F[{a,1+a/2,b,c,d,e},{a/2,1+a-b,1+a-c,1+a-d,1+a-e},-1]
                                a
                         a, 1 + -, b, c, d, e
                                2
Out[6]=  F                                                ; -1
        6 5  a
             -, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e
             2
In[7]:= %/.TListe
Out[7]= {}
In[8]:= Tgl7632
Do you want to set values for the equation?  [y|n]:  n
                                    a
                             a, 1 + -, b, c, d, e, -n
                                    2
Out[8]=  F                                                           ; 1   ==
        7 6  a
             -, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a + n
             2
                                                    (1 + a, 1 + a - d - e)
                1 + a - b - c, d, e, -n                                   n
⟩  F                                       ; 1   -----------------------
     4 3  1 + a - b, 1 + a - c, -a + d + e - n      (1 + a - d, 1 + a - e)
                                                                          n
In[9]:= Limes[%,n-⟩Infinity]

                                a
                         a, 1 + -, b, c, d, e
                                2
Out[9]=  F                                                ; -1   ==
        6 5  a
             -, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e
             2

          1 + a - b - c, d, e
      F                       ; 1 Γ(1 + a - d) Γ(1 + a - e)
     3 2  1 + a - b, 1 + a - c
⟩  --------------------------------------------------------
                    Γ(1 + a) Γ(1 + a - d - e)
In[10]:= F[{1+a/2,1,c,d,e,f},{a/2,1+a-c,1+a-d,1+a-e,1+a-f},1]
                              a
                          1 + -, 1, c, d, e, f
                              2
Out[10]=  F                                                ; 1
         6 5  a
              -, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a - f
              2
In[11]:= %/.TListe
Out[11]= {}
In[12]:= %%/.FEinf
Add the parameter:  a
                                 a
                          a, 1 + -, 1, c, d, e, f
                                 2
Out[12]=  F                                                   ; 1
         7 6     a
              a, -, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a - f
                 2
In[13]:= %/.TListe
Is -f a nonnegative integer?
[y|n]:    n
Is -1 - a + e + f a nonnegative integer?
[y|n]:    n
Is -1 - a + d + f a nonnegative integer?
[y|n]:    n
Is -1 - a + d + e a nonnegative integer?
[y|n]:    n

Is -1 - a + c + f a nonnegative integer?
[y|n]:    n
Is -1 - a + c + e a nonnegative integer?
[y|n]:    n
Is -1 - a + c + d a nonnegative integer?
[y|n]:    n
Is -a + f a nonnegative integer?
[y|n]:    n
Is -a + e a nonnegative integer?
[y|n]:    n
Is -a + d a nonnegative integer?
[y|n]:    n
Is -a + c a nonnegative integer?
[y|n]:    n
Be sure to apply "FOrdne" before using the following information!
Out[13]= {{T7631}, {T7634}, {T7635}, {T7691}, {T7692}}
\underline{See also}: SListe, FPerm, FTausche, TransListe.

trans

\underline{Description}: (a)_{n} → (−1)^{n} (1 − n − a)_{n}.

\underline{Usage}: Expr/.trans.

\underline{Example(s)}:

In[1]:= p[a,n]
Out[1]= (a)
           n
In[2]:= %/.trans
            n
Out[2]= (-1)  (1 - a - n)
                         n
\underline{See also}: Ers, PosListe, ManipulationsListe.

TransListe

\underline{Description}: List of all transformation formulas.

\underline{Usage}: TransListe.
\underline{See also}: TransListe$gl, TListe.

166 Alphabetic List: ZB

TransListe$gl \underline{Description}: List of all transformation formulas.

\underline{Usage}: TransListe$gl.
\underline{See also}: TransListe.

V \underline{Description}: V[a,List,z] is the very well-poised hypergeometric series.

\underline{Usage}: V[a,List,z].

\underline{Example(s)}:

In[1]:= V[a,{b,c},z]
                        a
                 a, 1 + -, b, c
                        2
Out[1]=  F                          ; z
        4 3  a
             -, 1 + a - b, 1 + a - c
             2
\underline{See also}: F, TeXFV, P, FFormat.

ZB \underline{Description}: Rule that looks for a recurrence relation for a sum or hypergeometric series using Zeilberger’s algorithm

[10, 11, 12]. Here a call is made to the function Zb of the Mathematica implementation of Gosper’s and Zeilberger’s algorithms written by Peter Paule and Markus Schorn. The current version 1.1 or updates can be received via e-mail request to peter.paule@risc.uni-linz.ac.at. This implemenation provides the user with the objects

    Zb, Gosper, RunMode, FileName, SolAmount, Fnk, GoRat, GoSol, Cert, DegBound, System, SystemDimension.

Also within the package HYP, all these objects work as described in the documentation of this implementation. Therefore the user is referred to this documentation and the description [5] in order to learn about the various features of these objects. The package HYP provides two additional objects, ZB and GOSPER. The rule ZB allows to apply Zeilberger’s algorithm directly to an expression containing a Sum or a hypergeometric series.

\underline{Usage}: Expr/.ZB[recvar, order].

\underline{Example(s)}:

In[1]:= Sum[Binomial[N,k]*Binomial[M,L-k],{k,0,Infinity}]

∞

             (        ) (   )
         \ (   M    ) ( N )
Out[1]= ⟩ (        ) (   )
         /   ( -k + L ) ( k )
             (        ) (   )
         k=0

In[2]:= %/.ZB[L,1]
    Peter Paule and Markus Schorn's implementation of the
             Zeilberger algorithm.  (Version 1.1)
Out[2]= {(-L + M + N) SUM[L] + (-1 - L) SUM[1 + L] == 0}
In[3]:= %1/.SumF

              -L, -N
         F            ; 1   (1 - L + M)
        2 1  1 - L + M                 L
Out[3]= --------------------------------
                      (1)
                         L
In[4]:= %/.ZB[L,1]
    Peter Paule and Markus Schorn's implementation of the
             Zeilberger algorithm.  (Version 1.1)
Is N a nonnegative integer?
[y|n]:    n
Out[4]= {(-L + M + N) SUM[L] + (-1 - L) SUM[1 + L] == 0}

\underline{See also}: GOSPER.

zerl

\underline{Description}: \begin{matrix}(a)_{n} → (a)_{m}(a + m)_{n−m}, \\ Γ(a) → (a + m)/(a)_{m}. \end{matrix}

The parameter m has to be entered on request.

\underline{Usage}: Expr/.zerl.

\underline{Example(s)}:

In[1]:= p[a,2*n]
Out[1]= (a)
           2 n
In[2]:= %/.zerl
bottom-split by:  m+n

Out[2]= (a)      (a + m + n)
           m + n            -m + n

\underline{See also}: Ers, PosListe, ManipulationsListe.

zus1

\underline{Description}: \begin{matrix}(a)_{n}(a + n)_{m} → (a)_{n+m}, \\ (a)_{n}/Γ(a + n) → 1/Γ(a). \end{matrix}

\underline{Usage}: Expr/.zus1.

\underline{Example(s)}:

In[1]:= p[a,2*n]*p[a+2*n,m-n]
Out[1]= (a)    (a + 2 n)
           2 n          m - n
In[2]:= %/.zus1
Out[2]= (a)
           m + n
In[3]:= p[a,2*n]/GAMMA[a+2*n]
          (a)
             2 n
Out[3]= ----------
        Γ(a + 2 n)
In[4]:= %/.zus1
         1
Out[4]= ----
        Γ(a)

\underline{See also}: zus2, zus3, erw1, erw2, Ers, PosListe, ManipulationsListe.

zus2

\underline{Description}: \begin{matrix}(a)_{n}/(a)_{m} → (a + m)_{n−m}, \\ Γ(a) (a)_{m} → Γ(a + m). \end{matrix}

\underline{Usage}: Expr/.zus2.

\underline{Example(s)}:

In[1]:= p[a,m]/p[a,n]*p[b,m+n]

        (a)  (b)
           m    m + n
Out[1]= -------------
            (a)
               n
In[2]:= %/.zus2
Out[2]= (b)      (a + n)
           m + n        m - n
In[3]:= p[a,m]*GAMMA[a]*p[b,m+n]
Out[3]= Γ(a) (a)  (b)
                m    m + n
In[4]:= %/.zus2
Out[4]= Γ(a + m) (b)
                    m + n
\underline{See also}: zus1, zus3, erw1, erw2, Ers, PosListe, ManipulationsListe.

zus3

\underline{Description}: (a)_{n}/(b)_{m} → (a)_{n−m},

provided a + n = b + m, and

Γ(a + n)/Γ(a) → (a)_{n}.

\underline{Usage}: Expr/.zus3.

\underline{Example(s)}:

In[1]:= p[a+m,n]/p[a+n,m]
        (a + m)
               n
Out[1]= --------
        (a + n)
               m
In[2]:= %/.zus3
Out[2]= (a + m)
               -m + n
In[3]:= GAMMA[a]/GAMMA[a+n]

         Γ(a)
Out[3]= --------
       Γ(a + n)

In[4]:= %/.zus3

Out[4]= (a + n)
              -n

In[5]:= %/.neg1

        1
Out[5]= ----
       (a)
          n

\underline{See also}: zus1, zus2, erw1, erw2, Ers, PosListe, ManipulationsListe.

References

1. W. N. Bailey, Generalized hypergeometric series, Cambridge University Press, Cambridge, 1935.

2. G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics And Its Applications 35, Cambridge University Press, Cambridge, 1990.

3. R. W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA 75 (1978), 40–42.

4. C. Krattenthaler, HYP and HYPQ — Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively q-binomial sums and basic hypergeometric series.

5. P. Paule and M. Schorn, A Mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities, J. Symbolic Comput. (to appear).

6. M. Rahman and A. Verma, Quadratic transformation formulas for basic hypergeometric series, Trans. Amer. Math. Soc. 335 (1993), 277–302.

7. L. J. Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966.

8. A. Verma and V. K. Jain, Some summation formulae for nonterminating basic hypergeometric series, SIAM J. Math. Anal. 16 (1985), 647–655.

9. S. Wolfram, MATHEMATICA — A system for doing mathematics by computer, second edition, Addison–Wesley, New York, 1991.

10. D. Zeilberger, A fast algorithm for proving terminating hypergeometric identities, Discrete Math. 80 (1990), 207–211.

11. D. Zeilberger, A holonomic systems approach to special functiions identities, J. Comput. Appl. Math. 32 (1990), 321–368.

12. D. Zeilberger, The method of creative telescoping, J. Symbolic Comput. 11 (1991), 195–204.