Combinatorial Identities Associated with a Multidimensional Polynomial Sequence

Isabel Cação and Helmuth R. Malonek CIDMA and Department of Mathematics University of Aveiro 3810-193 Aveiro Portugal

                        isabel.cacao@ua.pt
                          hrmalon@ua.pt

M. Irene Falcão CMAT and Department of Mathematics and Applications University of Minho 4710-057 Braga Portugal

                        mif@math.uminho.pt

Graça Tomaz CIDMA and Research Unit for Inland Development Department of Mathematics Polytechnic of Guarda 6300-659 Guarda Portugal

                          gtomaz@ipg.pt

Abstract

In this paper we combine the knowledge of different structures of a special Appell multidimensional polynomial sequence with the problem of establishing combinatorial identities. The elements of this special polynomial sequence have values in a Clifford algebra, are homogeneous hypercomplex differentiable functions of different degrees and their coefficients properties can be used to stress interesting matrix and combinatorial relations.

1 Motivation

The construction of polynomials with values in a Clifford algebra, based on Appell’s concept of power-like polynomials, has been considered in a series of articles by authors of this paper [1, 6, 7, 12, 13, 21]. The consideration of these polynomials was motivated by the fact that positive integer powers of the usually considered hypercomplex variable are not holomorphic, except for the complex case, while hypercomplex Appell polynomials are hyperholomorphic and have the behavior of power-like functions under hypercomplex differentiation. The study of sets of Appell polynomials became a focus of many authors attention, due to their theoretical importance and interesting applications [2, 3, 4, 5, 9, 11, 17, 18, 19, 25].

The knowledge of different hypercomplex polynomial bases systems can be combined with the problem of establishing and proving new combinatorial identities by bijective methods, linking in this way Clifford analysis and combinatorics. This idea was considered in the works [8, 10, 14, 23], illustrating that the use of the underlying noncommutative algebra can lead to generalizations of known results but, at the same time, opens the way for deriving new useful formulas and combinatorial identities.

In this work we pursued the idea of considering (non-standard) applications of Clifford algebras in the solution of problems of combinatorial nature, pointing out new combinatorial aspects of generalized Appell polynomials associated with two different representations. The paper is organized as follows: in Section 2 we introduce the necessary notation from Clifford analysis and describe briefly a set of homogeneous polynomials _{k}^{n} x generalizing the complex power function z^{k}. Two different representations of the polynomials _{k}^{n} x allow to derive, in Section 3, interesting matrix and combinatorial relations between their coefficients. In Section 4, a family of Pascal trapezoids closely related to the problem considered in the previous section is introduced and several patterns in such structure are studied.

2 A multidimensional polynomial sequence

Let e_{1}, e_{2}, …, e_{n} , n 2, be an orthonormal basis of the Euclidean vector space ℝ^{n} with a non-commutative product given according to the multiplication rules

e_{i}e_{j} e_{j}e_{i} 2δ_{ij}, i, j 1, 2, …, n,

where δ_{ij} is the Kronecker symbol. A basis for the associative 2^{n}-dimensional real Clifford algebra ℓ_{0,n} is the set e_{A} A 1, …, n with

e_{A} e_{h_{1}}e_{h_{2}} e_{h_{r}}, 1 h_{1} h_{r} n, e_{}e_{0} 1.

In general, the vector space ℝ^{n 1} is embedded in ℓ_{0,n} by identifying the element x_{0}, …, x_{n} in ℝ^{n 1} with the so-called paravector

x x_{0} _{k 1}^{n} e_{k}x_{k} x_{0} \underline{x} _{n} span_{ℝ} 1, e_{1}, …, e_{n} ℓ_{0,n}.

The conjugate x̄ and the norm x of x are given by x̄ x_{0} \underline{x} and x xx̄ ^{1 2} x̄x ^{1 2} _{k 0}^{n} x_{k}^{2} ^{1 2}.

The generalized Cauchy-Riemann operator in ℝ^{n 1} is defined by

\overline{∂} \frac{1}{2} ∂_{0} ∂\underline{x} ,

with ∂_{0}  \frac{∂}{∂x_{0}} and ∂\underline{x}  _{k 1}^{n} e_{k} \frac{∂}{∂x_{k}} . The conjugate generalized Cauchy-Riemann operator,

also called the hypercomplex differential operator, is denoted by

∂ \frac{1}{2} ∂_{0} ∂\underline{x} .

Functions defined in an open subset Ω ℝ^{n 1} _{n} with values in the Clifford algebra ℓ_{0,n} are of the form f z _{A} f_{A} z e_{A}, with f_{A} z real valued.

A function f is called left (right) monogenic in Ω if it is a solution of the differential equation \overline{∂}f 0 (f\overline{∂} 0).

The hypercomplex differentiability as generalization of complex differentiability has to be understood in the following way: a function f defined in an open domain Ω ℝ^{n 1} is hypercomplex differentiable if it has a uniquely defined areolar derivative f^{}in each point of Ω. Then, f is real differentiable and f^{}∂f. Furthermore, f is hypercomplex differentiable in Ω if and only if f is monogenic. In addition, the monogenicity of f implies that f^{}∂_{0}f ∂\underline{x}f (see [16, 20]).

Throughout this paper, a sequence of real numbers is denoted by a_{k} _{k} or simply a_{k} . A finite k-tuple of numbers can be seen as an infinite sequence by adding zeros to the end.

We consider particular polynomial sequences _{k}^{n} x _{k} whose terms are homogeneous monogenic polynomials of degree k, taking their values in _{n}, defined by

_{k}^{n} x _{s 0}^{k}T_{s}^{k} n x^{k s}x̄^{s},  (1)

where

T_{s}^{k} n {k \atop s} ^{\frac{\frac{n 1}{2} _{k s} \frac{n 1}{2} _{s}}{n _{k}}},  (2)

and a _{r} denotes the Pochhammer symbol, i.e., a _{r} \frac{Γ a r}{Γ a} , r 1, and a _{0} 1. Such polynomials were introduced initially [13, 21] as functions of a paravector variable x and its conjugate x̄ and the coefficients T_{s}^{k} n were calculated in such a way that _{k}^{n} x is a generalized Appell sequence of monogenic homogeneous polynomials with respect to ∂, i.e.,

1. _{0}^{n} x 1,

2. ∂ _{k}^{n} x k _{k 1}^{n} x , k 1, 2, ….

In the context of this paper, the representation of the Appell polynomials (1) in the form

_{k}^{n} x _{s 0}^{k} {k \atop s} c_{s} n x_{0}^{k s} \underline{x}^{s},  (3)

will play an important role. Such representation has been studied in a series of papers (see, e.g., [13, 14, 22]), where, in particular, it is proved that the coefficients are given by

c_{s} n  ;  ;  ;  ;  ;  ; \frac{s!! n 2 !!}{n s 1 !!} , if s is odd; c_{s 1} n , if s is even,  (4)

and satisfy

c_{k} n _{s 0}^{k} 1 ^{s}T_{s}^{k} n , k 0, 1, ….  (5)

In the relation (4), n!! denotes the usual double factorial of n, which (we recall) is defined as n!! n n 2 !! for n 1, and 0!! 1!! 1.

It is also worth mentioning that the numbers T_{s}^{k} n , for each degree k, constitute a partition of unity, i.e.,

1 _{s 0}^{k} T_{s}^{k} n , k 0, 1, …,  (6)

(cf. [14, Thm. 3.7]).

The family of numbers composed by T_{s}^{k} n , can be represented as a triangular table with lines of height k 0, 1, … and ordered from s 0 up to s k. Table 1 shows its first lines followed by the first terms of the sequences c_{k} n . Table 2 contains the first numbers associated with T_{s}^{k} 2 and T_{s}^{k} 3 , while in Table 3 we collect the first elements of the sequences c_{2k} n , (for n 2, …, 7). In the latter two tables one can see connections between c_{2k} n and T_{s}^{k} n with sequences from The On-Line Encyclopedia of Integer Sequences (OEIS) [26].

In the recent work [8], authors of this paper constructed a sequence m_{k} closely related to c_{k} 2 . Analyzing the sequence T_{0}^{k} 2 (cf. Table 2)

1, \frac{3}{4}, \frac{5}{8}, \frac{35}{64}, \frac{63}{128}, \frac{231}{512}, \frac{429}{1024}, \frac{6435}{16384}, \frac{12155}{32768}, \frac{46189}{131072}, \frac{88179}{262144}, …

one can see that it consists of rational numbers whose denominators are strictly increasing powers of 2. The sequence m_{k} of the corresponding exponents,

0, 2, 3, 6, 7, 9, 10, 14, 15, 17, 18, …

is called, in [8], the minimal exponent integer sequence with respect to 2, since it represents the least non-negative integer m_{k} such that 2^{m_{k}}T_{0}^{k} ℕ. The general expression of m_{k} reads as follows:

m_{k} k \frac{k 1}{2}  \frac{k 1}{2^{2}}  \frac{k 1}{2^{m}} ,

Table 1: The triangles T_{s}^{k} n and the sequences c_{k} n

T_{s}^{k} n 1

\frac{n 1}{2n}  \frac{n 1}{2n}
\frac{n 3}{4n}  \frac{n 1}{2n}  \frac{n 1}{4n}
\frac{n 5 n 3}{8n n 2}  \frac{3 n 3 n 1}{8n n 2}  \frac{3 n^{2} 1}{8n n 2}  \frac{n 3 n 1}{8n n 2}
\frac{n 5 n 7}{16n n 2}  \frac{n 5 n 1}{4n n 2}  \frac{3 n^{2} 1}{8n n 2}  \frac{n^{2} 1}{4n n 2}  \frac{n 5 n 1}{16n n 2}
\frac{n 5 n 7 n 9}{32n n 2 n 4}  \frac{5 n 7 n 5 n 1}{32n n 2 n 4}  \frac{5 n^{2} 1 n 5}{16n n 2 n 4}  \frac{5 n^{2} 1 n 3}{16n n 2 n 4}  \frac{5 n^{2} 1 n 5}{32n n 2 n 4}  \frac{n 5 n 1 n 7}{32n n 2}

c_{k} n 1, \frac{1}{n}, \frac{1}{n}, \frac{3}{n^{2} 2n} , \frac{3}{n^{2} 2n} , \frac{15}{n n^{2} 6n 8} , \frac{15}{n n^{2} 6n 8} , \frac{105}{n n 2 n 4 n 6} , \frac{105}{n n 2 n 4 n 6} , …

Table 2: First rows of T_{s}^{k} 2 and T_{s}^{k} 3 and the links to OEIS

n 2  n 3
\begin{matrix}1 \\ \frac{3}{4} & \frac{1}{4} \\ \frac{10}{16} & \frac{4}{16} & \frac{2}{16} \\ \frac{35}{64} & \frac{15}{64} & \frac{9}{64} & \frac{5}{64} \\ \frac{126}{256} & \frac{56}{256} & \frac{36}{256} & \frac{24}{256} & \frac{14}{256} \\  \end{matrix}\begin{matrix}1 \\ \frac{2}{3} & \frac{1}{3} \\ \frac{3}{6} & \frac{2}{6} & \frac{1}{6} \\ \frac{4}{10} & \frac{3}{10} & \frac{2}{10} & \frac{1}{10} \\ \frac{5}{15} & \frac{4}{15} & \frac{3}{15} & \frac{2}{15} & \frac{1}{15} \\  \end{matrix}

T_{s}^{k} 2 \frac{\underline{A078817} k k 1 2 s}{\underline{A000302} k} T_{s}^{k} 3 \frac{\underline{A004736} k k 1 2 s}{\underline{A026741} k 2}

Table 3: First elements of c_{2k} n , (n 2, …, 7) and the links to OEIS

\begin{aligned}c_{2k} 2 1, \frac{1}{2}, \frac{3}{8}, \frac{5}{16}, \frac{35}{128}, \frac{63}{256}, \frac{231}{1024}, … & =\frac{\underline{A001790} k}{\underline{A046161} k} \\ c_{2k} 3 1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \frac{1}{11}, \frac{1}{13}, … & =\frac{1}{\underline{A005408} k} \\ c_{2k} 4 1, \frac{1}{4}, \frac{1}{8}, \frac{5}{64}, \frac{7}{128}, \frac{21}{512}, \frac{33}{1024}, … & =\frac{\underline{A098597} k}{\underline{A120777} k} \\ c_{2k} 5 \frac{3}{3}, \frac{3}{15}, \frac{3}{35}, \frac{3}{63}, \frac{3}{99}, \frac{3}{143}, \frac{3}{195}, … & =\frac{3}{\underline{A000466} k 1} \\ c_{2k} 6 1, \frac{1}{6}, \frac{1}{16}, \frac{1}{32}, \frac{7}{384}, \frac{3}{256}, \frac{33}{4096}, … & =\frac{\underline{A099398} k}{\underline{A099399} k} \\ c_{2k} 7 \frac{5}{5}, \frac{5}{35}, \frac{5}{105}, \frac{5}{231}, \frac{5}{429}, \frac{5}{715}, \frac{5}{1105}, … & =\frac{5}{\underline{A162540} k} \end{aligned}

with m log_{2} k 1 , (see [8, Thm. 6] for details). The sequence, now listed in OEIS as \underline{A283208}, allows to write

T_{0}^{k} 2 2c_{2k 1} 2 \frac{\underline{A001790} k 1}{2^{\underline{A283208} k}} .

3 Connection-like identities

Connection-like is here understood as the problem of linking the coefficients T_{s}^{k} n and c_{s} n associated with the two representations (1) and (3) of the same polynomial sequence _{k}^{n} x . In this section we describe bijections between those coefficients and present corresponding matrix relations.

We let T_{k} n and C_{k} n denote the k 1 -dimensional vectors

T_{k} n T_{0}^{k} n T_{1}^{k} n T_{k 1}^{k} n T_{k}^{k} n ^{T}

and

C_{k} n c_{0} n c_{1} n c_{k 1} n c_{k} n ^{T} .

We start with a technical result relating the numbers (2) in the row k i with their distant neighbors in the row k.

Lemma 1. For any fixed value of k 1, the following relation holds:

T_{s}^{k i} n _{j 0}^{i} A k, i, s, j T_{s j}^{k} n ; i 1, …, k, s 0, …, k i,  (7)

where

A k, i, s, j {i \atop j} \begin{matrix}k i s 1 & _{i j} & s 1 & _{j} \\ k i 1 & _{i} \end{matrix} .  (8)

Proof. The result is true for i 1, since (7) reads as

T_{s}^{k 1} n \frac{k s}{k} T_{s}^{k} n \frac{s 1}{k} T_{s 1}^{k} n , s 0, …, k 1.  (9)

This relation between an element in row k 1 and two adjacent elements in the row k was already proved in [14, Thm. 3.5].

The repeated use of (9) allows to express an element in row k i in terms of i 1 adjacent elements in the row k. In fact, for s 0, …, k 2 one can write

T_{s}^{k 2} n \frac{k s 1}{k 1} T_{s}^{k 1} n \frac{s 1}{k 1} T_{s 1}^{k 1} n
\frac{k s 1}{k 1} \frac{k s}{k} T_{s}^{k} n \frac{s 1}{k} T_{s 1}^{k} n \frac{s 1}{k 1} \frac{k s 1}{k} T_{s 1}^{k} n \frac{s 2}{k} T_{s 2}^{k} n
\frac{k s 1 _{2}}{k 1 _{2}} T_{s}^{k} n \frac{2 k s 1 s 1}{k 1 _{2}} T_{s 1}^{k} n \frac{s 1 _{2}}{k 1 _{2}} T_{s 2}^{k} n .

Applying the same reasoning we have

T_{s}^{k 3} n \frac{k s 2 _{3}}{k 2 _{3}} T_{s}^{k} n \frac{3 k s 2 _{2} s 1}{k 2 _{3}} T_{s 1}^{k} n \frac{3 k s 2 s 1 _{2}}{k 2 _{3}} T_{s 2}^{k} n \frac{s 1 _{3}}{k 2 _{3}} T_{s 3}^{k} n

T_{s}^{k i} n _{j 0}^{i} _{j}^{i} \frac{k s i 1 _{i j} 1 s _{j}}{k i 1 _{i}} T_{s j}^{k} n

and the result is proved.

The next result is a generalization to arbitrary dimensions of Theorem 2 in [21] (see also [12]).

Theorem 2. The components T_{s}^{k} n and T_{s}^{k 1} n of the vectors T_{k} n and T_{k 1} n , respectively, satisfy the k 1 k 1 system of algebraic equations represented in matrix form as

M_{k}T_{k} n ; ; ; ; ; ; ; ; ; \begin{matrix}kT_{k 1} n \\ c_{k} n \end{matrix} ; ; ; ; ; ; ; ; ; ,

where M_{k} is given by

M_{k}   ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}k & 1 & 0 & 0 &  & 0 & 0 \\ 0 & k 1 & 2 & 0 &  & 0 & 0 \\ 0 & 0 & k 2 & 3 &  & 0 & 0 \\  &  &  &  &  &  &  \\ 0 & 0 & 0 & 0 &  & k 1 & 0 \\ 0 & 0 & 0 & 0 &  & 1 & k \\ 1 & 1 & 1 & 1 &  & 1 ^{k 1} & 1 ^{k} \end{matrix}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; , k 1.  (10)

Proof. It is a simple consequence of relation (9) together with the alternating sum formula (5) for c_{k} n .

Lemma 3. The matrix M_{k} k 1, 2, … in (10) is nonsingular and

det M_{k} 2 ^{k}k!.

Proof. Developing the determinant along the last row, we obtain

det M_{k} 1 ^{k 2}k! 1 ^{k 3}k \frac{k!}{1!} 1 ^{k 4}k k 1 \frac{k!}{2!} 1 ^{k 5}k k 1 k 2 \frac{k!}{3!}
1 ^{k}k k 1 k 2 1\frac{k!}{k!}
1 ^{k}k! 1 \frac{k}{1!} \frac{k k 1}{2!} \frac{k k 1 k 2}{3!}  \frac{k!}{k!}
1 ^{k}k! _{j 0}^{k} _{j}^{k}  1 ^{k}k!2^{k}.

Remark 4. We point out that det M_{k} 1 ^{k}\underline{A000165} k .

As a consequence of the previous results one can derive recursive matrix relations between the vectors T_{k} n and C_{k} n .

Theorem 5. The values of T_{s}^{k} n and c_{s} n , k 0, 1, …, s 0, 1, …, k, are related by the following matrix relation

T_{k} n N_{k}C_{k} n ,  (11)

where N_{k} is recursively defined by

N_{0} 1
N_{k} M_{k}^{1}  ;  ;  ;  ;  ; \begin{matrix}kN_{k 1} & 0 \\ 0 & 1 \end{matrix}  ;  ;  ;  ;  ; , k 1, 2, ….  (12)

Moreover, if we denote by { \atop N_{k}} the matrix N_{k}^{1} then we have

C_{k} n { \atop N_{k}}T_{k} n ,  (13)

where

{ \atop N_{0}} 1
{ \atop N_{k}}  ;  ;  ;  ;  ; \begin{matrix}\frac{1}{k} { \atop N_{k 1}} & 0 \\ 0 & 1 \end{matrix}  ;  ;  ;  ;  ; M_{k}, k 1, 2, ….  (14)

Proof. To prove the relations (11)-(12) we use an inductive process on k. Taking into account Theorem 2 and the fact that T_{0}^{0} n 1 c_{0} n , one can write, for k 1,

T_{1} n M_{1}^{1} \begin{matrix}T_{0} n \\ c_{1} n \end{matrix} M_{1}^{1} 1 0 0 1 \begin{matrix}c_{0} n \\ c_{1} n \end{matrix} N_{1}C_{1} n ,

with N_{1} M_{1}^{1} \begin{matrix}1N_{0} & 0 \\ 0 & 1 \end{matrix} .

Now assume that T_{k} n N_{k}C_{k} n holds for k. For k 1 one has

T_{k 1} n M_{k 1}^{1} \begin{matrix}k 1 T_{k} n \\ c_{k 1} n \end{matrix} M_{k 1}^{1} \begin{matrix}k 1 N_{k}C_{k} n \\ c_{k 1} n \end{matrix}

M_{k 1}^{1} \begin{matrix}k 1 N_{k} & 0 \\ 0 & 1 \end{matrix} \begin{matrix}C_{k} n \\ c_{k 1} n \end{matrix} N_{k 1}C_{k 1} n ,

with N_{k 1} M_{k 1}^{1} \begin{matrix}k 1 N_{k} & 0 \\ 0 & 1 \end{matrix} .

To prove the relations (13)-(14), observe that N_{k} is invertible, since

det N_{k} k^{k} det M_{k}^{1} det N_{k 1} \frac{k^{k}}{k! 2 ^{k}} det N_{k 1}
\frac{k}{2} ^{k} \frac{1}{k!} \frac{k 1}{2} ^{k 1} \frac{1}{k 1 !} det N_{k 2}  _{m 1}^{k} \frac{m}{2} ^{m} \frac{1}{m!}.

The inversion of N_{k} leads to the desired result.

The next result allows to express each component of the vector T_{k} n as a linear combination of c_{0} n , …, c_{k} n .

Theorem 6. The numbers T_{s}^{k} n , k 0, 1, …, s 0, 1, …, k, can be written as

T_{s}^{k} n \frac{1}{2^{k}} {k \atop s} _{j 0}^{k} σ_{s,j}^{k}c_{j} n ,  (15)

where

σ_{s,j}^{k} _{m 0}^{s} 1 ^{m} {s \atop m} {k s \atop j m} .  (16)

Proof. Replacing x_{0} \frac{x x̄}{2} and \underline{x} \frac{x x̄}{2} in (3) we obtain successively^{1}

_{k}^{n} x \frac{1}{2^{k}} _{j 0}^{k} {k \atop j} c_{j} n x x̄ ^{k j} x x̄ ^{j}

\frac{1}{2^{k}} _{j 0}^{k} {k \atop j} c_{j} n { \atop }_{t 0}^{k j} {k j \atop t} x^{k j t} x̄^{t}{ \atop }{ \atop }_{l 0}^{j} 1 ^{l} {j \atop l} x^{j l}x̄^{l}{ \atop }

\frac{1}{2^{k}} _{j 0}^{k} {k \atop j} c_{j} n _{s 0}^{k} _{m 0}^{s} 1 ^{s m} {k j \atop m} {j \atop s m} x^{k s}x̄^{s}

\frac{1}{2^{k}} _{j 0}^{k} {k \atop j} c_{j} n _{s 0}^{k} _{m 0}^{s} 1 ^{m} {k j \atop s m} {j \atop m} x^{k s}x̄^{s}.

By comparing last expression with (1) we obtain

T_{s}^{k} n \frac{1}{2^{k}} _{j 0}^{k} {k \atop j} c_{j} n _{m 0}^{s} 1 ^{m} {k j \atop s m} {j \atop m} .

The result follows, since _{j}^{k} _{s m}^{k j} _{m}^{j} _{s}^{k} _{m}^{s} _{j m}^{k s} .

The explicit expression of each c_{j} n , j 0, …, k, as a linear combination of T_{s}^{k} n , s 0, …, k, reads as follows.

Theorem 7. For any fixed value of k, the coefficients c_{k i} n can be written as

c_{k i} n \frac{1}{_{i}^{k}} _{s 0}^{k} 1 ^{s}σ_{s,i}^{k}T_{s}^{k} n , i 0, 1, …, k.  (17)

Proof. Using Lemma 1 together with (5) we obtain

c_{k i} n _{s 0}^{k i} 1 ^{s}T_{s}^{k i} n _{s 0}^{k i} 1 ^{s} _{j 0}^{i} A k, i, s, j T_{s j}^{k}

_{j 0}^{i} A k, i, 0, j T_{j}^{k} _{j 1}^{i} A k, i, 1, j 1 T_{j}^{k} 1 ^{k i} _{j k i}^{k} A k, i, k i, j k i T_{j}^{k}.

Observe that if we allow in the expression (8) of A k, i, s, j s and j to be arbitrary integers, then A k, i, s, j vanishes whenever j i or j 0 or k s i 1. Therefore we can write

c_{k i} n _{j 0}^{k} A k, i, 0, j T_{j}^{k} _{j 0}^{k} A k, i, 1, j 1 T_{j}^{k}  1 ^{k i} _{j 0}^{k} A k, i, k i, j k i T_{j}^{k}

_{s 0}^{k} _{m 0}^{k i} 1 ^{m}A k, i, m, s m T_{s}^{k} n

1For convenience we include binomial coefficients _{k}^{n} for nonnegative integer n and integer k such that k 0 or k n, which as usually are considered to be zero.

or, equivalently

c_{k i} n ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; \begin{matrix}_{s 0}^{k} _{m 0}^{s} 1 ^{m}A k, i, m, s m _{m s 1}^{k i} 1 ^{m}A k, i, m, s m T_{s}^{k} n , & if k i s; \\ _{s 0}^{k} _{m 0}^{s} 1 ^{m}A k, i, m, s m _{m k i 1}^{s} 1 ^{m}A k, i, m, s m T_{s}^{k} n , & if k i s. \end{matrix}

Since A k, i, m, s m 0, for m s, the sum _{m s 1}^{k i} 1 ^{m}A k, i, m, s m is zero. On the other hand, we also have A k, i, m, s m 0 in _{m k i 1}^{s} 1 ^{m}A k, i, m, s m , since k m i 1. We have proved that

c_{k i} n _{s 0}^{k} ϑ_{i,s}^{k}T_{s}^{k} n , i 0, …, k,

where

ϑ_{i,s}^{k} _{m 0}^{s} 1 ^{m}A k, i, m, s m , i 0, …, k.

The use of (8) allows to write

ϑ_{i,s}^{k} _{m 0}^{s} 1 ^{m} \frac{i!}{i s m ! s m !} \frac{k i m 1 ... k s m 1 ...s}{k i 1 ...k}

_{m 0}^{s} 1 ^{m} \frac{i! k i !}{k!} \frac{k s !}{i s m ! k i m !} \frac{s!}{s m !m!} \frac{1 ^{s}}{_{i}^{k}} σ_{s,i}^{k}

(cf. (16)) and the result follows.

Remark 8. Since σ_{s,k}^{k} 1 ^{s}, Theorem 7 contains as a particular case (i k) the already known property (6).

Theorem 5 can be used together with Theorems 6 and 7 to obtain the main result of this section: the explicit form of the matrix that relates vectors T_{k} n and C_{k} n .

Theorem 9. Consider the coefficients σ_{s,j}^{k} introduced in (16). Then the following relations are true:

i) T_{k} n N_{k}C_{k} n , where N_{k} is the k 1 k 1 matrix whose elements n_{ij}^{k} are

n_{ij}^{k} \frac{1}{2^{k}} {k \atop i 1} σ_{i 1,j 1}^{k} , i, j 1, …, k 1.

ii) C_{k} n { \atop N_{k}}T_{k} n , where { \atop N_{k}} is the k 1 k 1 matrix whose elements { \atop n_{ij}^{k}} are

{ \atop n_{ij}^{k}} \frac{1 ^{j 1}}{_{k i 1}^{k}} σ_{j 1,k i 1}^{k} , i, j 1, …, k 1.

Proof. Observe that Theorem 6 can be expressed in matrix form as

T_{k} n \frac{1}{2^{k}} S_{k}C_{k} n ,

where S_{k} is the k 1 k 1 matrix whose elements s_{ij} are such that

s_{ij} {k \atop i 1} σ_{i 1,j 1}^{k} , i, j 1, …, k 1.

Result i) follows at once from (11), while result ii) is a simple consequence of Theorem 7.

The last theorem points out that the connection between T_{k} n and C_{k} n is provided by the matrices N_{k} and { \atop N_{k}}. However, a deeper observation permits to recognize that such connection is actually done by the coefficients σ_{s,j}^{k}, since they are the common factors in the coefficients of linear combinations (15)-(17). Examples of matrices N_{k}, { \atop N_{k}}, and vectors T_{k} n , C_{k} n , k 1, …, 4, are given in Table 4.

4 A family of Pascal trapezoids

In addition to providing the link between T_{s}^{k} n and c_{s} n , the coefficients σ_{i,j}^{k} (cf. (16)) in the identities (15) and (17) have several other interesting properties. In fact, considering i, j, k arbitrary nonnegative integers such that j k, the numbers σ_{i,j}^{k} can be represented, for each fixed i, as a triangle with rows k (k 0, 1, …) and ordered from j 0 to j k, as illustrated in Table 5. This section is dedicated to the study of various patterns in such structure and to the discussion of properties and combinatorial identities.

We observe that

1. for i 0, we obtain the Pascal triangle, since σ_{0,j}^{k} _{j}^{k} , (see first triangle in Table 5);

2. for i 1, σ_{1,j}^{k}  _{j}^{k 1}  _{j 1}^{k 1}  \frac{k 2j}{k} _{j}^{k} , which corresponds to the Catalan triangle

numbers C_{k,j} \frac{k 2j}{k} _{j}^{k} mentioned by Miana et al. [24].

In general, for each fixed value of i, σ_{i,j}^{k} can be obtained recursively, as stated in the following result.

Theorem 10 (Pascal recurrence). For each fixed value of i, the numbers σ_{i,j}^{k} satisfy the following linear recurrence relation

σ_{i,j 1}^{k 1} σ_{i,j}^{k} σ_{i,j 1}^{k}, 0 j k 1, k i  (18)

Table 4: The matrices N_{k}, { \atop N_{k}} N_{k}^{1} , and the vectors T_{k} n , C_{k} n , k 1, …, 4
k  N_{k}  { \atop N_{k}}  T_{k}  C_{k}

1  \frac{1}{2}  ;  ;  ;  ;  ; \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \frac{n 1}{2n} ; \frac{n 1}{2n}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; 1_{\frac{1}{n}}  ;  ;  ;  ;  ;

2  \frac{1}{2^{2}}  ;  ;  ;  ;  ;  ;  ; \begin{matrix}1 & 2 & 1 \\ 2 & 0 & 2 \\ 1 & 2 & 1 \end{matrix}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \end{matrix}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \frac{n 3}{4n} ; \frac{n 1}{4n} ; \frac{n 1}{2n}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; 1 ; \frac{1}{n} ; \frac{1}{n}  ;  ;  ;  ;  ;  ;  ;

3  \frac{1}{2^{3}}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}1 & 3 & 3 & 1 \\ 3 & 3 & 3 & 3 \\ 3 & 3 & 3 & 3 \\ 1 & 3 & 3 & 1 \end{matrix}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}1 & 1 & 1 & 1 \\ 1 & \frac{1}{3} & \frac{1}{3} & 1 \\ 1 & \frac{1}{3} & \frac{1}{3} & 1 \\ 1 & 1 & 1 & 1 \end{matrix}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \frac{3 n 5 n}{8n 2 n} ; \frac{3 1 n 3 n}{8n 2 n} ; \frac{3 1 n 1 n}{8n 2 n} ; \frac{1 n 3 n}{8n 2 n}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; 1 ; \frac{1}{n} ; \frac{1}{n} ; \frac{3}{n n 2}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;

\frac{1}{2^{4}}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}1 & 4 & 6 & 4 & 1 \\ 4 & 8 & 0 & 8 & 4 \\ 6 & 0 & 12 & 0 & 6 \\ 4 & 8 & 0 & 8 & 4 \\ 1 & 4 & 6 & 4 & 1 \end{matrix}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}1 & 1 & 1 & 1 & 1 \\ 1 & \frac{1}{2} & 0 & \frac{1}{2} & 1 \\ 1 & 0 & \frac{1}{3} & 0 & 1 \\ 1 & \frac{1}{2} & 0 & \frac{1}{2} & 1 \\ 1 & 1 & 1 & 1 & 1 \end{matrix}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \frac{n 5 n 7}{16n n 2} ; \frac{n 1 n 5}{4n n 2} ; \frac{3 n 1 n 1}{8n n 2} ; \frac{n 1 n 1}{4n n 2} ; \frac{n 1 n 5}{16n n 2}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; 1 ; \frac{1}{n} ; \frac{1}{n} ; \frac{3}{n n 2} ; \frac{3}{n n 2}  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;

with boundary conditions

σ_{i,0}^{k} 1,  σ_{i,k}^{k} 1 ^{i}, k i  (19)

and initial values

σ_{i,j}^{i}  {i \atop j} 1 ^{j}, j 1, …, i 1.  (20)

Proof. When k i and 0 j k 1 we immediately obtain from the definition of σ_{i,j}^{k} that (cf. (16))

σ_{i,j}^{k} σ_{i,j 1}^{k} _{m 0}^{i} 1 ^{i m} _{m}^{i}  _{j m}^{k i}  _{j 1 m}^{k i}  _{m 0}^{i} 1 ^{i m} _{m}^{i} _{j 1 m}^{k i 1} σ_{i,j 1}^{k 1}.

When j 0 the sum in the right-hand side of (16) reduces to the case m 0 whereas when j k the corresponding sum reduces to the case m i. Therefore

σ_{i,0}^{k} 1 and σ_{i,k}^{k} 1 ^{i}.

Table 5: Triangles associated with σ_{i,j}^{k}, i 0, …, 3

[figure: 1; 1; 1; 1; 2; 1; 1; 3; 3; 1; 1; 4; 6; 4; 1; 1; 5; 10; 10; 5; 1; 1; 6; 15; 20; 15; 6; 1; 1; 7; 21; 35; 35; 21; 7; 1; ; ; ; ; ; ; ; ; i 0; 0; 1; -1; 1; 0; -1; 1; 1; 1; -1; 1; 2; 0; 2; -1; 1; 3; 2; 2; 3; -1; 1; 4; 5; 0; 5; 4; -1; 1; 5; 9; 5; 5; 9; 5; -1; ; ; ; ; ; ; ; ; i 1; 2; 2; 2; 1; -2; 1; 1; 1; 1; 1; 1; 0; 2; 0; 1; 1; 1; 2; 2; 1; 1; 1; 2; 1; 4; 1; 2; 1; 1; 3; 1; 5; 5; 1; 3; 1; ; ; ; ; ; ; ; ; i 2; 0; 6; 6; 6; 6; 6; 1; -3; 3; -1; 1; 2; 0; 2; -1; 1; 1; 2; 2; 1; -1; 1; 0; 3; 0; 3; 0; -1; 1; 1; 3; 3; 3; 3; 1; -1; ; ; ; ; ; ; ; ; i 3]

On the other hand, if k i the same sum only has the term corresponding to m j. Hence

σ_{i,j}^{i} {i \atop j} 1 ^{j}, j 0, …, i.

Theorem 10 supports the idea that the above recurrence relation together with the boundary conditions (19) and initial conditions (20) lead, for each i, to a Pascal trapezoid which can be seen as a substructure of the triangle array σ_{i,j}^{k}. In Table 5, we can observe the trapezoids corresponding to the particular cases of i 1, i 2, and i 3 as well as the boundary and initial conditions. These trapezoids are related to the sequences \underline{A037012}, \underline{A182533}, and \underline{A230206}.

In the context of this paper, we use the designation of Pascal trapezoid of order i 1, 2, … to refer to the trapezoidal array whose entries are σ_{i,j}^{k}, k i, 0 j k. The following properties can easily be derived. Property 11.

(i) Pascal trapezoids of even order are symmetric, i.e., σ_{2i,j}^{k} σ_{2i,k j}^{k}; (ii) Pascal trapezoids of odd order are anti-symmetric, i.e., σ_{2i 1,j}^{k} σ_{2i 1,k j}^{k} ; (iii) The Hockey-stick identity is valid, i.e., _{r j}^{k} σ_{i,j}^{r} σ_{i,j 1}^{k 1}. Proof. From (16) we have

σ_{i,k j}^{k} _{m 0}^{i} 1 ^{m} _{m}^{i} _{k j m}^{k i}  _{m 0}^{i} 1 ^{m} _{m}^{i} _{j m i}^{k i}  _{m 0}^{i} 1 ^{i m} _{m}^{i} _{j m}^{k i}  1 ^{i}σ_{i,j}^{k}

and (i) and (ii) are proved. According to (18)-(19),

_{r j}^{k} σ_{i,j}^{r} σ_{i,j}^{j} _{r j 1}^{k} σ_{i,j}^{r} 1 ^{i} _{r j 1}^{k} σ_{i,j 1}^{r 1} σ_{i,j 1}^{r}

_{r j 1}^{k} σ_{i,j 1}^{r 1} _{r j 1}^{k 1} σ_{i,j 1}^{r 1} σ_{i,j 1}^{k 1},

and the proof of (iii) is complete. Property 12. The central coefficients of the Pascal trapezoid of order i are linked to the central binomial coefficients in the following way:

σ_{i,k}^{2k} ; ; ; ; ; ; ; ; ; ; 0, if i is odd; 1 ^{r} \begin{matrix} & {k \atop r} & \\ & {2k \atop 2r} & \end{matrix}{2k \atop k} , if i 2r is even.

Proof. Observe that

σ_{i,k}^{2k} _{m 0}^{i} 1 ^{m} {i \atop m} {2k i \atop k m} \begin{matrix} & {2k \atop k} & \\ & {2k \atop i} & \end{matrix}_{m 0}^{i} 1 ^{m} {k \atop m} {k \atop i m}

and the result follows by using the identity [15, Formula (1.19)]

_{m 0}^{i} 1 ^{m} {k \atop m} {k \atop i m} ; ; ; ; ; ; ; ; 0, if i is odd; 1 ^{\frac{i}{2}} k_{\frac{i}{2}} , if i is even.

It is worth noting that the odd case also follows from Theorem 6.

Observe that σ_{i,j}^{k} can also be written as (cf. (16))

σ_{i,j}^{k} _{m 0}^{j} 1 ^{m} {i \atop m} {k i \atop j m} ,

since _{j m}^{k i} 0, for m j, whereas _{m}^{i} 0, for m i. This leads to the conclusion that the entries of the Pascal trapezoid of order i are nothing else than the coefficients in the expansion of 1 x ^{i} 1 x ^{k i}, i.e.,

1 x ^{i} 1 x ^{k i} _{j 0}^{k} σ_{i,j}^{k}x^{j}  (21)

which means that the family of Pascal trapezoids also contains as particular cases the sequences \underline{A230207} up to \underline{A230212}. Moreover, one can deduce the following properties.

Property 13.

(i) Sum of the rows: _{j 0}^{k} σ_{i,j}^{k} 0, for k i 0;

(ii) Alternating sum of the rows: _{j 0}^{k} 1 ^{j}σ_{i,j}^{k} 0, for k i;

(iii) Chu-Vandermonde identity: _{m 0}^{j} σ_{i,m}^{k_{1}}σ_{i,j m}^{k_{2}} σ_{2i,j}^{k_{1} k_{2}} ;

(iv) Sum of the squares of the rows: _{j 0}^{k} σ_{i,j}^{k} ^{2} 1 ^{i}σ_{2i,k}^{2k}.

Proof. The first two properties follow at once by considering x 1 and x 1 in (21). The Chu-Vandermonde formula follows by noting that

1 x ^{i} 1 x ^{k_{1} i} 1 x ^{i} 1 x ^{k_{2} i} 1 x ^{2i} 1 x ^{k_{1} k_{2} 2i}

and equating the coefficients in

_{j 0}^{k_{1}} σ_{i,j}^{k_{1}}x^{j} _{j 0}^{k_{2}} σ_{i,j}^{k_{2}}x^{j} _{j 0}^{k_{1} k_{2}} σ_{2i,j}^{k_{1} k_{2}} x^{j}.

Property (iv) follows from (iii) and using Property 11 (i) or (ii).

5 Acknowledgments

The work of the third author was supported by Portuguese funds through the CMAT — Centre of Mathematics and FCT within the Project UID/MAT/00013/2013. The work of the other authors was supported by Portuguese funds through the CIDMA — Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEst-OE/MAT/UI4106/2013.

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2010 Mathematics Subject Classification: Primary 05A10; Secondary 05A19, 11B83, 30G35. Keywords: binomial coefficient, special sequence, hypercomplex Appell polynomial.

(Concerned with sequences \underline{A000165}, \underline{A000302}, \underline{A000466}, \underline{A001790}, \underline{A004736}, \underline{A005408}, \underline{A026741}, \underline{A037012}, \underline{A046161}, \underline{A078817}, \underline{A098597}, \underline{A099398}, \underline{A099399}, \underline{A120777}, \underline{A162540}, \underline{A182533}, \underline{A230206}, \underline{A230207}, \underline{A230208}, \underline{A230209}, \underline{A230210}, \underline{A230211}, \underline{A230212}, and \underline{A283208}.)

Received July 21 2018; revised version received August 6 2018. Published in Journal of Integer Sequences, August 23 2018.

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