Riordan Arrays, Sheffer Sequences and “Orthogonal” Polynomials

Giacomo Della Riccia

Dept. of Math. and Comp. Science - Research Center Norbert Wiener University of Udine Via delle Scienze 206 33100 Udine Italy

                        dlrca@uniud.it

Abstract

Riordan group concepts are combined with the basic properties of convolution families of polynomials and Sheffer sequences, to establish a duality law, canonical forms ρ(n, m) = \binom{n}{m}c^{m}F_{n−m}(m), c ≠ 0, and extensions ρ(x, x − k) = (−1)^{k}x\underline{k+1}c^{x−k}F_{k}(x), where the F_{k}(x) are polynomials in x, holding for each ρ(n, m) in a Riordan array. Examples ρ(n, m) = \binom{n}{m}S_{k}(x) are given, in which the S_{k}(x) are “orthogonal” polynomials currently found in mathematical physics and combinatorial analysis.

1 Introduction

We derive from basic principles in the theory of convolution families [10] and Sheffer sequences [16] of polynomials, canonical forms ρ(n, m) = \binom{n}{m}c^{m}F_{n−m}(m), c ≠ 0, and extensions ρ(x, x − k) = x\underline{k}c^{x−k}F_{k}(x − k)/k! = (−1)^{k}x\underline{k+1}c^{x−k}ρ_{k}(x), holding for all elements ρ in the Riordan group. We show in Section 3 that the extensions are of polynomial type when c = 1. In Section 2, we define transformation rules and a duality law, that will greatly simplify algebraic manipulations. In Section 4, we give examples ρ(n, m) = \binom{n}{m}S_{n−m}(m), in which S_{k}(x) are “orthogonal” polynomials currently found in mathematical physics and combinatorial analysis. The concluding remarks are in Section 5.

For easy reference, we recall some definitions. The Riordan group is a set of invertible infinite lower triangular matrices M = {ρ(n, m)}_{n,m≥0}, called Riordan matrices, with entries:

ρ(n, m) = [\frac{u^{n}}{n!} ] g(u)\frac{f(u)^{m}}{m!} = \frac{n!}{m!}[u^{n−m}]g(u) (\frac{f(u)}{u})^{m} , ρ(0, 0) = 1,  (1)
equivalently, ∑_{n≥m≥0} ρ(n, m)\frac{u^{n}}{n!} = g(u)\frac{f(u)^{m}}{m!} (“exponential” Riordan group),

where g(u) = 1 + g_{1}u + g_{2}u^{2} + … is invertible, with ĝ(u) = 1/g(u) = ∑_{n≥0} ĝ_{n}u^{n}, and f(u) = f_{1}u + f_{2}u^{2} + … is a delta series with compositional inverse \overline{f}(u), f(\overline{f}(u)) = \overline{f}(f(u)) = u. A group element is denoted by ρ = (g(u), f(u)) and the group law ⋆ is (l(u), h(u)) ⋆ (g(u), f(u)) = (l(u)g(h(u)), f(h(u)), with identity I = (1, u) and group inverse ρ^{−1} = (1/g(\overline{f}(u)), \overline{f}(u)). A group representation ρ = ρ_{1} ⋆ ρ_{2}, in matrix notation, is ρ(n, m) = ∑_{i} ρ_{1}(n, i)ρ_{2}(i, m). As we know, group representations are extensively used in the study of identities [13, 18, 14, 15] and they will play an essential role in this work. A sequence a_{k}(x), given by ∑_{k} a_{k}(x)u^{k} = a(u)^{x}, is a convolution family of polynomials iff a(0) = 1 [10]. In a convolution family the polynomials a_{k}(x) are multiples of x for k > 0, having degree ≤ k, and xa_{0}(x) ≡ 1. A Sheffer sequence S_{k}(x) for (g(u), f(u)) [16] is given by ∑_{k≥0} S_{k}(x)u^{k}/k! = (1/g(\overline{f}(u)))e^{x\overline{f}(u)}, where (g(u), f(u)) are as in Riordan group theory; the Sheffer sequence for f = (1, f(u)) is the associated sequence for f(u), and the Sheffer sequence for (g(u), u) is the Appell sequence for g(u).

2 Transformation Rules and the Duality Law

We first define some useful transformation rules, noting that transformation rules and Barry’s “transforms” [2] are different concepts.

1) The duality rule “∼” for a function f(n, m), n, m integers, is f̃(n, m) = f(−m, −n) and for a function f_{k}(x), the rule is f̃_{k}(x) = f_{k}(k − x). When x = n and k = n − m, the two rules coincide. The following special cases will be used in the sequel, without reference.

\binom{ñ}{m} = \begin{matrix} \\ ( & {n \atop n − m} & ) \end{matrix} = \binom{−m}{n − m} = (−1)^{n−m}\binom{n − 1}{m − 1}, n ≥ m ≥ 0,

˜\frac{n!}{m!} = { \atop n\underline{n−m}} = (−m)\underline{n−m} = (−1)^{n−m}(n − 1)\underline{n−m} = (−1)^{n−m} \frac{n!}{m!} \frac{m}{n},

; \frac{n!}{(m − 1)!} = { \atop n\underline{n−m+1}} = (−1)^{n−m+1} \frac{n!}{(m − 1)!}.

The dual ρ̃(n, m) of ρ(n, m) and the dual of a group representation ρ = ρ_{1} ⋆ ρ_{2} are

ρ̃(n, m) = ρ(−m, −n) = (−1)^{n−m} \frac{n!}{m!} \frac{m}{n}[u^{n−m}]g(u) (\frac{f(u)}{u})^{−n} from (1),

ρ̃(n, m) = ∑_{i} ρ_{1}(−m, −i)ρ_{2}(−i, −n) = ∑_{i} ρ̃_{2}(n, i)ρ̃_{1}(i, m), ρ̃ = ρ̃_{2} ⋆ ρ̃_{1},

after a change i → −i; this change is legal because we are allowed to use summation ∑_{i} over an unbounded range.

2) For any number μ, the scaling rule “(μ)” is a group automorphism:

(μ)ρ = (1, μu) ⋆ ρ ⋆ (1, \frac{u}{μ}) = (g(μu), \frac{f(μu)}{μ}) , lim_{μ→0}(μ)(g(u), f(u)) = (1, u) = I,

((μ)ρ)(n, m) = μ^{n−m}ρ(n, m), ((μ)ρ)^{−1} = (μ)ρ^{−1}, (μ)(ρ_{1} ⋆ ρ_{2}) = (μ)ρ_{1} ⋆ (μ)ρ_{2}.

3) The negation rule “(-)” is the special scaling μ = −1:

(−)ρ = (g(−u), −f(−u)) : ((−)ρ)(n, m) = (−1)^{n−m}ρ(n, m).

Duality and negation applied to (1) and Lagrange’s inversion formula yield, respectively,

(−1)^{n−m}f̃(n, m) = (−1)^{n−m}f(−m, −n) = \frac{n!}{m!} \frac{m}{n}[u^{n−m}] (\frac{f(u)}{u})^{−n} ,
f^{−1}(n, m) = \frac{n!}{m!}[u^{n}]\overline{f}(u)^{m} = \frac{n!}{m!} \frac{m}{n}[u^{n−m}] (\frac{f(u)}{u})^{−n} .

Combining the two formulas, we find a duality law in the associated subgroup {(1, f(u))}:

f^{−1}(n, m) = (−1)^{n−m}f̃(n, m), f^{−1} = (−)f̃,  (2)

For g in the Appell subgroup {(g(u), u)} and ρ^{−1} = f^{−1} ⋆ g^{−1} = (−)f̃⋆ g^{−1}, (1) yields:

g(n, m) = \frac{n!}{m!}g_{n−m} = \binom{n}{m}(n − m)!g_{n−m}, g^{−1}(n, m) = \binom{n}{m}(n − m)!ĝ_{n−m},

ρ^{−1}(n, m) = ∑_{i=0}^{n−m}(−1)^{i}f̃(n, n − i)(n − i)\underline{n−m−i}ĝ_{n−m−i}.

3 Canonical Forms and Extensions

Let us write the delta series f(u) in f = (1, f(u)) and the defining relation (1) as:

f(u) = uca(u) = uc(1 + a_{1}u + ⋯), c ≠ 0,
ρ(n, m) = \binom{n}{m}c^{m} [\frac{u^{n−m}}{(n − m)!}] g(u) (\frac{f(u)}{cu})^{m} = \binom{n}{m}c^{m}F_{n−m}(m),  (3)

and call c ≠ 0 the weight of f, \{ca_{n}\} the f-reference sequence (f -refseq) and \binom{n}{m}c^{m}F_{n−m}(m) the canonical form of ρ(n, m). The remarkable structure of canonical forms, namely, a binomial coefficient multiplied by a factor c^{m}F_{n−m}(m), implies the following.

Proposition 1. A numerical array with entries ρ(n, m) = \binom{n}{m}c^{m}F_{n−m}(m) is a Riordan matrix iff the F_{k}(x) are polynomials forming a Sheffer sequence.

Proof. If the ρ(n, m) are numbers related to an element ρ = (g(u), f(u)), then according to (3) we can write

∑_{n−m} \frac{F_{n−m}(m)}{(n − m)!}u^{n−m} = g(u)e^{m ln(\frac{f(u)}{cu})} = g(u) (\frac{f(u)}{cu})^{m} ,

thus the F_{k}(x) form a Sheffer sequence given by ∑_{k} F_{k}(x)u^{k}/k! = g(u)e^{x ln(f(u)/cu)}. Conversely, if the above Sheffer sequence is given, then, by letting x = m, we obtain (3), proving that the ρ(n, m) define a Riordan matrix.

Corollary 2. Each ρ(n, m) in a Riordan matrix has an extension

ρ(x, x − k) = (−1)^{k}x\underline{k+1}ρ_{k}(x), ρ_{k}(x) = c^{x−k} \frac{(−1)^{k}}{k!} \frac{F_{k}(x − k)}{x − k},  (4)

which is of polynomial type when c = 1.

Proof. Since F_{n−m}(m) is a polynomial in m, we can put in the canonical form n − m = k, n = x and m = x − k, thus obtaining immediately well-defined extensions (4), which are clearly of polynomial type when c = 1.

Since f(u)/cu = a(u) is invertible and a(0) = 1, the coefficients (−1)^{k}xf_{k}(x + k) in the power series expansion of (f(u)/cu)^{x}, written as (f(u)/cu)^{x} = x ∑_{k}(−1)^{k}f_{k}(x + k)u^{k}, form a convolution family. Similarly, (1) written for f^{−1}(n, m) = (−1)^{n−m}f̃(n, m), with u → cu and m = −x, implies:

(\frac{c\overline{f}(u)}{u})^{m} = −m ∑_{n−m≥0}(−1)^{n−m}f_{n−m}(−m)\frac{u^{n−m}}{c^{n−m}} ,

(\frac{\overline{f}(cu)}{u})^{−x} = (\frac{\overline{f(u)/c}}{u})^{−x} = x ∑_{k≥0}(−1)^{k}f_{k}(x)u^{k}, c\overline{f} (f (\frac{u}{c})) = u,

showing that the (−1)^{k+1}xf_{k}(−x) form a convolution family for (\overline{f(u)/c}/u)^{x}. Moreover,

f(n, m) = (−1)^{n−m} \frac{n!}{(m − 1)!}c^{m}f_{n−m}(n),
f^{−1}(n, m) = (−1)^{n−m}f̃(n, m) = (−1)^{n−m+1} \frac{n!}{(m − 1)!}c^{−n}f_{n−m}(−m)
f(x, x − k) = c^{x−k}(−1)^{k}x\underline{k+1}f_{k}(x), f^{−1}(x, x − k) = c^{−x}(−1)^{k+1}x\underline{k+1}f_{k}(k − x).

The f_{k}(x), k > 0, will be called the f -polynomial sequence (f -polseq). One can verify that if f has weight c, then f^{−1} has weight 1/c, and that changes of scale keep weights invariant, hence, they cannot modify the characteristics of an extension.

For g ∈ {(g(u), u)}, we have simply:

g(n, m) = \binom{n}{n}(n − m)!g_{n−m}, g^{−1}(n, m) = \binom{n}{n}(n − m)!ĝ_{n−m},
g(x, x − k) = g_{k}x\underline{k}, g^{−1}(x, x − k) = ĝ_{k}x\underline{k}.

For ρ(n, m), in addition to (3) and (4), and for ρ^{−1}(n, m) we write relations that, when g(u) ≡ 1, reduce to the corresponding expressions for f(n, m) and f^{−1}(n, m):

ρ(n, m) = (−1)^{n−m} \frac{n!}{(m − 1)!}ρ_{n−m}(n),
ρ^{−1}(n, m) = \binom{n}{m}c^{−n}G_{n−m}(m) = (−1)^{n−m+1} \frac{n!}{(m − 1)!}ρ_{n−m}^{−1}(−m),
ρ^{−1}(x, x − k) = (−1)^{k+1}x\underline{k+1}ρ_{k}^{−1}(k − x), ρ_{k}^{−1}(x) = c^{−x} \frac{(−1)^{k}}{k!} \frac{G_{k}(−x)}{x}.

When the nonzero entries in a f-refseq are of the same sign, we say that ρ = (g(u), f(u)) is of the second kind, otherwise, of the first kind. In a pair {ρ, ρ^{−1}}, at most one element can be of the 2nd kind; when such an element exists, it will be denoted ρ^{−1}. Similarly, capital letters in an inverse pair {ϕ, Φ} will, in general, indicate elements of the 2nd kind.

Duality defines a dual element ρ̃ that extends ρ(n, m) to all integers n, m, and since, as we have seen, ρ̃(n, m), ρ(n, m) and ρ^{−1}(n, m) are tied together, it is natural to include these numbers in a single extended ρ-array that will represent the pair {ρ, ρ^{−1}}.

We adopt the term generalized numbers for the ρ(n, m) in the sense that these numbers extend to all integer values n, m,. The ρ-refseq: c ∑_{i=0}^{n} g_{i}a_{n−i} given by

\frac{ρ(n + 1, 1)}{(n + 1)!} = c[u^{n}]g(u)\frac{f(u)}{cu} = c[u^{n}]g(u)a(u) = c ∑_{i=0}^{n} g_{i}a_{n−i}.

4 Riordan Arrays and “Orthogonal” Polynomials

Consider a Sheffer sequence S_{k}(x) for (g(u), f(u)) and the elements U = (1/g(\overline{f}(u)), ue^{\overline{f}(u)})

and C = (1/g(\overline{f}(u)), \overline{f}(u)), then we can write:

∑_{k≥0} \frac{S_{k}(x)}{k!}u^{k} = \frac{e^{x\overline{f}(u)}}{g(\overline{f}(u))} = ∑_{i≥0} x^{i} \frac{1}{i!} \frac{\overline{f}(u)^{i}}{g(\overline{f}(u))} = ∑_{i≥0} x^{i} ∑_{k≥0} C(k, i)\frac{u^{k}}{k!} ,

S_{k}(x) = ∑_{i=0}^{k} C(k, i)x^{i}, C(n, m) = \binom{n}{m} [\frac{u^{n−m}}{(n − m)!}] \frac{(\frac{\overline{f}(u)}{u})m}{g(\overline{f}(u))} ,

U(n, m) = \binom{n}{m} [\frac{u^{n−m}}{(n − m)!}] \frac{e^{m\overline{f}(u)}}{g(\overline{f}(u))} = \binom{n}{m}S_{n−m}(m) = \frac{(−1)^{n−m+1}n!}{(m − 1)!}U_{n−m}(−m),

U(x, x − k) = (−1)^{k+1}x\underline{k+1}U_{k}(k − x), U_{k}(x) = \frac{(−1)^{k}}{k!} \frac{S_{k}(−x)}{x}, k > 0, U_{0}(x) = \frac{1}{x}.

An immediate consequence of these equations is the following important relationship between Riordan group elements and Sheffer sequences.

Proposition 3. S_{k}(x) = ∑_{i=0}^{k} C(k, i)x^{i} are polynomials forming a Sheffer sequence for (g(u), f(u)) iff the coefficients C(n, m) are generalized numbers for C = (1/g(\overline{f}(u)), \overline{f}(u)).

We now give examples where the S_{k}(x) are classical “orthogonal” polynomials, currently found in mathematical physics and combinatorial analysis.

Example 1. In the typical \underline{Stirling-array} \{s = (1, ln(1 + u)), S = (1, e^{u} − 1)\}, we have:

s(n, m) = (−1)^{n−m} \frac{n!}{(m − 1)!}σ_{n−m}(n), S(n, m) = (−1)^{n−m+1} \frac{n!}{(m − 1)!}σ_{n−m}(−m);

s(x, x − k) = (−1)^{k}x\underline{k+1}σ_{k}(x), S(x, x − k) = (−1)^{k+1}x\underline{k+1}σ_{k}(k − x);

Stirling-polseq : σ_{k}(x); s-refseq : \frac{(−1)^{n}}{n + 1} , S-refseq : \frac{1}{(n + 1)!};

(\frac{ln(1 + u)}{u})^{x} = x ∑_{k≥0}(−1)^{k}σ_{k}(x + k)u^{k}, (\frac{e^{u} − 1}{u})^{−x} = x ∑_{k≥0}(−1)^{k}σ_{k}(x)u^{k}.

The \underline{Stirling-array} is related to the Exponential Polynomials ϕ_{k}(x) [12, p. 63] forming the associated sequence for f(u) = ln(1 + u), \overline{f}(u) = e^{u} − 1, γ^{−1} = (1, e^{u} − 1):

∑_{k} \frac{ϕ_{k}(x)}{k!}u^{k} = e^{x\overline{f}(u)} = e^{x(e^{u}−1)}, C(n, m) = S(n, m) : Stirling numbers (2d kind),

Φ = U = (1, ue^{e^{u}−1}) , ϕ_{k}(x) = ∑_{i=0}^{k} S(k, i)x^{i} : Touchard polynomials;

Φ(n, m) = \binom{n}{m}ϕ_{n−m}(m) = \frac{(−1)^{n−m+1}n!}{(m − 1)!}Φ_{n−m}(−m), Φ_{k}(x) = \frac{(−1)^{k}}{k!} \frac{ϕ_{k}(−x)}{x};

ϕ_{k}(1) = ϖ_{n} = ∑_{i=0}^{k} S(k, i) : Bell numbers, given by e^{e^{u}−1} = ∑_{k} ϖ_{n}u^{k}/k!.

The Iterated Exponential Polynomials ϕ_{k}^{[q]}(x) form the associated sequence for f(u) = s^{[q]}(u) = s(s^{[q−1]}(u)), s^{[0]} = u; \overline{f}(u) = S^{[q]}(u) = S(S^{[q−1]}(u)), S^{[0]} = u:

∑_{k} \frac{ϕ_{k}^{[q]}(x)}{k!}u^{k} = e^{xS^{[q]}}, ϕ_{k}^{[q]}(x) = ∑_{i=0}^{k} S^{[q]}(k, i)x^{i}; S^{[q]} = \underbrace{S ⋆ S ⋆ … ⋆ S}_{q terms},
Φ^{[q]} = U^{[q]} = (1, ue^{S^{[q]}(u)}) , Φ^{[1]} = Φ, C^{[q]} = (1, S^{[q]}(u))
S^{[q]}(n, m) = \binom{n}{m}ϕ_{n−m}^{[q]}(m), Φ_{k}^{[q]}(x) = \frac{(−1)^{k}}{k!} \frac{ϕ_{k}^{[q]}(−x)}{x}.

The \underline{Stirling-array} corresponds to q = 1, and for q = 2, we have the \underline{Stir^{[2]}-array} \{β = (1, β(u)), B}, β(u) = ln(1 + ln(1 + u)), B(u) = e^{e^{u}−1} − 1, B-refseq : ϖ_{n+1}/(n + 1)!,

β(n, m) = ∑_{i=m}^{n} s(n, i)s(i, m), B(n, m) = ∑_{i=m}^{n} S(n, i)S(i, m),

∑_{k} \frac{ϕ_{k}^{[2]}(x)}{k!}u^{k} = e^{x ln(1+ln(1+u))} = (1 + ln(1 + u))^{x}, ϕ_{k}^{[2]}(x) = ∑_{j≥i≥0}^{k} s(k, j)s(j, i)x^{i},

Stir^{[2]}-polseq : σ_{k}^{[2]}(x) = ∑_{i=0}^{k}(x − i)σ_{k−i}(x − i)σ_{i}(x).

Example 2. For the \underline{Lah-array} \{λ, Λ = (1, Λ(u))\}^{1}, Λ(u) = −λ(−u) = u/(1 − u/2):

Λ(n, m) = \frac{(−1)^{n−m+1}n!}{(m − 1)!}Lah_{n−m}(−m) = \binom{n}{m}\frac{(n − m)!}{2^{n−m}} \binom{n − 1}{m − 1} : scaled Lah numbers,

and for the \underline{Pascal-array} \{p, P = (1/(1 − u), u/(1 − u))\}, p = P^{−1} = (−)P:

P (n, m) = \frac{(−1)^{n−m+1}n!}{(m − 1)!}P as_{k}(−m) = \frac{n!}{m!}[u^{n−m}]\frac{1}{(1 − u)^{m+1}} = \binom{n}{m}\frac{n!}{m!}, by [8, (5.56)],

P ascal-polseq : P ascal_{k}(x) = \frac{(−1)^{k}}{k!} (x − 1)\underline{k−1}.

The above arrays are related to the Laguerre Polynomials L_{k}^{(a)}(x) of order a [12, p. 31, p. 108] forming the Sheffer sequence for (g(u) = (1 + u)^{−a−1}, f(u) = u/(u − 1) = \overline{f}(u)), given by ∑_{k} L_{k}^{(a)}(x)u^{k}/k! = (1 − u)^{−a−1}e^{xu/(u−1)}; L_{k}^{{0}}(x): (simple) Laguerre polynomials.

Lag^{(a)} = U = ((1 − u)^{−a−1}, ue^{\overline{f}(u)}), C^{(a)} = ((1 − u)^{−a−1}, \frac{u}{u − 1}) ,
Lag^{(a)}(n, m) = \binom{n}{m}L_{n−m}^{\{a\}}(m) = (−1)^{n−m+1} \frac{n!}{(m − 1)!}Lag_{n−m}^{(a)}(−m),
C^{(a)}(n, m) = (−1)^{m} \frac{n!}{m!}[u^{n−m}](1 − u)^{−a−1−m} = (−1)^{m} \frac{n!}{m!}\binom{a + n}{n − m}, by [8, (5.56)],

L_{k}^{(a)}(x) = ∑_{i=0}^{k} C^{(a)}(k, i)x^{i} = ∑_{i=0}^{k} \frac{k!}{i!}\binom{a + k}{k − i}(−x)^{i}.  (5)

The \underline{Lah-array} corresponds to a = −1 and the \underline{Pascal-array} to a = 0.

1Erratum: in Della Riccia [5, p. 3, line before last]: Λ(u) = u/(1 + u/2) should be Λ(u) = u/(1 − u/2).

Example 3. Let us consider the \underline{Tanh-array} \{θ = (1, θ(u)), Θ = (1, Θ(u))\}, with

θ(u) = 2\frac{e^{u} − 1}{e^{u} + 1} = 2 tanh \frac{u}{2}, Θ(u) = ln \frac{1 + u/2}{1 − u/2} = 2 arg tanh \frac{u}{2}, Θ = Λ ⋆ s, S = Λ ⋆ θ :

θ(n, m) = (−1)^{n−m} \frac{n!}{(m − 1)!}δ_{n−m}(n), Θ(n, m) = (−1)^{n−m+1} \frac{n!}{(m − 1)!}δ_{n−m}(−m),

δ_{k}(x) = ∑_{i=0}^{k} \frac{(−1)^{i+1}}{2^{i}} \frac{(k − x − 1)\underline{i}}{i!} σ_{k−i}(k − x − i), σ_{k}(x) = − ∑_{i=0}^{k} \frac{(x − 1)\underline{i}}{2^{i}i!} δ_{k−i}(k − x),

T anh-polseq : δ_{k}(x).

The \underline{Tanh-array} and the \underline{Tangent-array} \{arctan = (1, arctan u), tan = (1, tan u)\} are related by the elementary trigonometric formulas:

tan u = \frac{1}{ı} tanh(ıu) = \frac{1}{2ı}2 tanh (2ı\frac{u}{2}) = \frac{θ(2ıu)}{2ı}, arctan u = \frac{1}{ı} arg tanh(ıu) = \frac{Θ(2ıu)}{2ı},

which look like a scaling of θ and Θ with μ = 2ı, ı^{2} = −1: tan(n, m) = (2ı)^{n−m}θ(n, m), arctan(n, m) = (2ı)^{n−m}Θ(n, m). The numbers tan(n, m) and arctan(n, m) appear in Comtet [4, p.p.259-260] as T (n, k) and t(n, k). Putting T (n, m) = (2ı)^{n−m}(n!/(m − 1)!)δ_{n−m}(n) in the recursion for T (n, k) and after common factors are divided out, we are left with

(x + 1)δ_{k}(x + 1) = (x − k)δ_{k}(x) − \frac{x − k + 2}{4}δ_{k−2}(x), δ_{k}(x) ≡ 0, k < 0, xδ_{0}(x) ≡ 1,

(compare with(x + 1)σ_{k}(x + 1) = (x − k)σ_{k}(x) + xσ_{k−1}(x) [8, Exercise 6.18].  (6)

One can prove by induction that the δ_{k}(x), k = 2j > 0, have degree j − 1 and δ_{k}(x) ≡ 0 when k is odd; hence factors (−1)^{k} may be omitted, leaving us with simplified formulas:

(\frac{2 tanh \frac{u}{2}}{u} )^{x} = x ∑_{k≥0} δ_{k}(x + k)u^{k}, (\frac{1}{u} ln \frac{1 + \frac{u}{2}}{1 − \frac{u}{2}} )^{−x} = x ∑_{k≥0} δ_{k}(x)u^{k}; Θ = θ̃.

The \underline{Tanh-array} is related to the Mittag-Leffler Polynomials M_{k}(x) [12, p.75] forming the associated sequence for

f(u) = \frac{e^{u} − 1}{e^{u} + 1} = tanh \frac{u}{2}, \overline{f}(u) = ln \frac{1 + u}{1 − u} = 2 arg tanh u, ∑_{k} \frac{M_{k}(x)}{k!}u^{k} = (\frac{1 + u}{1 − u})^{x} .

MiLef = U = (1, u\frac{1 + u}{1 − u}) , C = (1, 2 arg tanh u), MiLef_{k}(x) = \frac{(−1)^{k}}{k!} \frac{M_{k}(−x)}{x},
MiLef(n, m) = \binom{n}{m}M_{n−m}(m) = \frac{(−1)^{n−m+1}n!}{(m − 1)!}MiLef_{n−m}(−m),

M_{k}(x) = 2^{k} ∑_{i=0}^{k} Θ(k, i)x^{i} = 2^{k} ∑_{i=0}^{k} ∑_{j} Λ(k, j))s(j, i)x^{i} = ∑_{j=0}^{k} \frac{k!}{j!}\binom{k − 1}{k − j}2^{j}x\underline{j},  (7)

where we used the identity ∑_{i} s(j, i)x^{i} = x\underline{j}.

Example 4. The simple \underline{Binom-array} \{bin = (exp^{−u}, u), Bin\},

bin(n, m) = (−1)^{n−m}\binom{n}{m}, Bin(n, m) = \binom{n}{m}, Bin(x, x − k) = (−1)^{k}bin(x, x − k) = \frac{x\underline{k}}{k!} ,

is related to the Poisson-Charlier Polynomials c_{k}^{{a}}(x) = a^{−k}L_{k}^{{x−k}}(a) [12, p.119], since the egf of c_{k}^{{a}}(0) = a^{−k}L_{k}^{{−k}}(a) = (−1)^{k} is e^{−u}, by using L_{k}^{{−k}}(a) = (−1)^{k}a^{k} from (5). The \underline{Tree-array} \{r = (1, ue^{−u}), R = (1, \overline{ue^{−u}})\},

r(n, m) = (−1)^{n−m}\binom{n}{m}m^{n−m}, R(n, m) = \binom{n}{m}\frac{m}{n}n^{n−m},
(\frac{r(u)}{u})^{x} = e^{−xu}, (\frac{R(u)}{u})^{−x} = x ∑_{k} \frac{(−1)^{k}}{k!} (x − k)^{k−1}u^{k}, T ree_{k}(x) = \frac{1}{k!}(x − k)^{k−1},
R(n, 1) = n^{n−1} : number of rooted trees of n vertices [4, p.152].

can be related to the case a = −1 of the Abel Polynomials A_{k}(x; a), a ≠ 0, [12, p.73], forming the associated sequence for f(u) = ue^{au}; \overline{f}(u) = \overline{ue^{au}}, A_{k}(x; a) = x(x − ak)^{k−1}.

Abel^{(a)} = U^{(a)} = (1, ue\overline{ue^{au}}), Abel_{k}^{(a)}(x) = \frac{(−1)^{k}}{k!} \frac{A_{k}(−x; a)}{x}, C^{(a)} = (1, \overline{ue^{au}});

Abel^{(a)}(n, m) = (−1)^{n−m+1} \frac{n!}{(m − 1)!}Abel_{n−m}^{(a)}(−m) = \binom{n}{m}A_{n−m}(m; a).

For a = −1, Abel_{k}^{(−1)}(x) = (−1)^{k} ∑_{i=0}^{k} \binom{k}{i}i^{k−i}(−x)^{i}, \binom{k}{i}i^{k−i} : idempotent numbers.

Example 5. The \underline{Sinh-array} \{argshin = (1, 2 arg sinh(u/2)/b), shin = (1, 2 sinh(bu/2))\} is related to the Gould Polynomials G_{k}(x; a, b), b ≠ 0, [12, p.67] forming the associated sequence for f(u) = e^{au}(e^{bu} − 1), b ≠ 0. In fact, when a = −b/2, G_{k}(x; b) = G_{k}(x; −b/2, b), f(u) = e^{bu/2} − e^{−bu/2} = 2 sinh(bu/2), c = b, \overline{f}(u) = 2 arg sinh(u/2)/b,

∑_{k} G_{k}(x; b)u^{k}/k! = e^{x(2/b) arg sinh(u/2)} = B_{1/2}(u)^{x/b} = (u/2 + \sqrt{1 + u^{2}/4})^{2x/b} [10, p.71],

where B_{1/2}(u) = (u/2 + \sqrt{1 + u^{2}/4})^{2} is a generalized binomial series [8, p.203].

Gould^{(b)} = U^{(b)} = (1, ue^{^{\frac{2}{b}} arg sinh ^{\frac{u}{2}}} ) , Gould_{k}^{(b)}(x) = \frac{(−1)^{k}}{k!} \frac{G_{k}(−x; b)}{x},
Gould^{(b)}(n, m) = (−1)^{n−m+1} \frac{n!}{(m − 1)!}Gould_{n−m}^{(b)}(−m) = \binom{n}{m}G_{n−m}(m; b);
G_{k}(x; 1) = x (x + \frac{1}{2}k − 1))\underline{k−1} : central factorial polynomials [12, p.68].

Example 6. The Bernoulli Polynomials B_{k}^{(a)}(x) of order a, a ≠ 0, [12, p.93] form the Appell sequence for g(u) = ((e^{u} − 1)/u)^{a}, given by ∑_{k} B_{k}^{(a)}(x)u^{k}/k! = (u/(e^{u} − 1))^{a}e^{xu}. B_{k}^{(m)} = B_{k}^{(m)}(0) are the higher order Bernoulli numbers and B_{k} = B_{k}^{(1)} are the Bernoulli numbers given by ∑_{k} B_{k}u^{k}/k! = u/(e^{u} − 1) = B(u).

Bern^{(a)} = U = ((\frac{u}{e^{u} − 1})^{a} , ue^{u}) , Bern^{(a)}(n, m) = \binom{n}{m}B_{n−m}^{(a)}(m)

C^{(a)} = ((\frac{u}{e^{u} − 1})^{a} , u) , C^{(a)}(n, m) = \binom{n}{m}B_{n−m}^{(a)},

B_{k}^{(a)}(x) = ∑_{i=0}^{k} \binom{k}{i}B_{k−i}^{(a)}x^{i}, B_{k}(x) = ∑_{i=0}^{k} \binom{k}{i}B_{k−i}x^{i} : Bernoulli polynomials.

The Nörlund polynomials B_{k}^{(x)} = B_{k}^{(x)}(0) [1] form the associated sequence for \overline{ln B(u)},

∑_{k} \frac{B_{k}^{(x)}}{k!} u^{k} = e^{x ln B(u)} = (\frac{e^{u} − 1}{u})^{−x} = x ∑_{k≥0}(−1)^{k}σ_{k}(x)u^{k},

\frac{B_{k}^{(x)}}{xk!} = (−1)^{k}σ_{k}(x) = \frac{s(x, x − k)}{x\underline{k+1}}, \frac{B_{k}^{(−x)}}{xk!} = (−1)^{k+1}σ_{k}(−x) = \frac{S(x + k, x)}{(x + k)\underline{k+1}}, (8)
\frac{B_{k}}{k!} = σ_{k}(1) − [k = 1] = −kσ_{k}(0), k > 0, by the recursion (6).  (9)

For the \underline{Bern-array} \{ber, Ber = (1, uB(u))\}, Ber-refseq: B_{n}/n!, we have

Ber(n, m) = \binom{n}{m}B_{n−m}^{(m)}, Bern_{k}(x) = \frac{(−1)^{k}}{k!} \frac{B_{k}^{(−x)}}{x} = −σ_{k}(−x), k ≥ 0.

In passing, we remark that the associated sequence U_{k}(x) for f(u) = \overline{uB(u)} is given by

∑_{k} \frac{U_{k}(x)}{k!}u^{k} = e^{xuB(u)} = ∑_{i} x^{i} \frac{(uB(u))^{i}}{i!} , U_{k}(x) = ∑_{i=0}^{k} \binom{k}{i}B_{k−i}^{(i)}x^{i}.

Now consider the Stirling polynomials convolution formula [8, (6.46)], written with t = 1, i → i − m, r → r + m and s → s + m:

−(r + m)(s + n) ∑_{i=m}^{n} σ_{i−m}(r + i)σ_{n−i}(−s − i) = (r − s + m − n)σ_{n−m}(r − s).

Using (8), we can write σ_{n−m}(r−s), σ_{i−m}(r+i) and σ_{n−i}(−s−i) in terms of Stirling numbers and Nörlund polynomials; after substitution in the convolution relation, we get

∑_{i=m}^{n} \frac{s(r + i, r + m)}{(r + i)\underline{i−m}} \frac{S(s + n, s + i)}{(s + n)\underline{n−i+1}} = \frac{r − s + m − n}{s + n} \frac{B_{n−m}^{(r−s)}}{(r − s)(n − m)!}.

For a = r − s, integers r, s ≥ 0, \frac{1}{(r + i)\underline{i−m}} = \frac{(r + m)!}{(r + i)!}, \frac{1}{(s + n)\underline{n−i+1}} = \frac{(s + i − 1)!}{(s + n)!} :

∑_{i=m}^{n} s(r + i, r + m)S(s + n, s + i)\frac{(s + i − 1)\underline{i−1}}{(r + i)!} = \frac{a + m − n}{s + n} \frac{(s + n)!}{(r + m)!} \frac{B_{n−m}^{(a)}}{a(n − m)!}.

With r = s, a = 0, we get from (8): and (9),

∑_{i=m}^{n} s(r + i, r + m)S(r + n, r + i)\frac{1}{i} = \frac{(r + n)\underline{n−m}}{r + n} (\frac{B_{n−m}}{(n − m)!} + [m = n − 1]) , m > 0,

which is a generalization of the case r = s = 0, that corresponds to the known identity:

∑_{i=m}^{n} s(i, m)S(n, i)\frac{1}{i} = \frac{1}{n}\binom{n}{m}\frac{B_{n−m}}{(n − m)!} + [m = n − 1], m > 0 [8, (6.100)],  (10)

The identity (10), in turn, generalizes the identity B_{n} = ∑_{i=0}^{n}(−1)^{i}i!S(n, i)/(i+1) appearing in Comtet [4, p.220] and which is the special case m = 1 of (10) and, at the same time, of Kaneko’s identity B_{n−m} = (−1)^{n−m+i}S(n − m, i)i!/(i + 1)^{m} [9, Theorem 1].

Example 7. The Euler Polynomials E_{k}^{(a)}(x) of order a, a ≠ 0, [12, p.100] form the Appell sequence for g(u) = ((e^{u} + 1)/2)^{a},

∑_{k} \frac{E_{k}^{(a)}(x)}{k!}u^{k} = (\frac{2}{e^{u} + 1})^{a} e^{xu} = (\frac{e^{− \frac{u}{2}}}{cosh \frac{u}{2}} )^{a} e^{xu}, E_{k}(x) = E_{k}^{(1)}(x) : Euler polynomials,

for u → 2u, x = \frac{a}{2} : ∑_{k} \frac{2^{k}E_{k}^{(a)}(\frac{a}{2})}{k!}u^{k} = (\frac{1}{cosh u})^{a} , E_{k} = 2^{k}E_{k} (\frac{1}{2}) : Euler numbers.

Euler^{(a)} = U = ((\frac{2}{e^{u} + 1})^{a} , ue^{u}) , Euler^{(a)}(n, m) = \binom{n}{m}E_{n−m}^{(a)}(m),

C^{(a)} = ((\frac{2}{e^{u} + 1})^{a} , u) , C^{(a)}(n, m) = \binom{n}{m}E_{n−m}^{(a)}(0),

E_{k}^{(a)}(x) = ∑_{i=0}^{k} C^{(a)}(k, i)x^{i} = ∑_{i=0}^{k} \binom{k}{i}E_{k−i}^{(a)}(0)x^{i}.

For the \underline{Euler-array} \{eul, Eul = (1, uE(u))\}, E(u) = 1/ cosh u, Eul-refseq: E_{n}/n!, we find

Eul(n, m) = \frac{n!}{m!}[u^{n−m}]E(u)^{m} = \binom{n}{m}2^{n−m}E_{n−m}^{(m)} (\frac{m}{2})

2^{n−m}E_{n−m}^{(m)} (\frac{m}{2}) = ∑_{i=0}^{n−m} \binom{n − m}{i}2^{i}m^{i}E_{i}^{(m)}(0), \frac{Eul(n + 1, 1)}{n + 1} = E_{n} = ∑_{i=0}^{n} \binom{n}{i}2^{i}E_{i}(0).

The egf of Euler polynomials evaluated at x = 0 and the gf G(u) for the Genocchi numbers G_{n} are related by

∑_{n} \frac{E_{n}(0)}{n!}u^{n} = (1 − tanh \frac{u}{2}) = \frac{2}{e^{u} + 1} = ∑_{n≥0} \frac{G_{n+1}}{(n + 1)!}u^{n} = \frac{G(u)}{u} [4, p.49],

thus, for the \underline{Geno-array} \{gen, Gen = (1, G(u))\}, Gen-refseq: G_{n+1}/(n + 1)!, we have

Gen(n, m) = \frac{n!}{m!}[u^{n}] (\frac{G(u)}{u})^{m} = \binom{n}{m}E_{n−m}^{(m)}(0), Gen_{k}(x) = \frac{(−1)^{k}}{k!} \frac{E_{k}^{(−x)}}{x} .

With G(u) = u (1 − tanh \frac{u}{2}) = u (1 − \frac{θ(u)}{2}) and \frac{θ(u)^{i}}{i!} = ∑_{j} θ(j, i)\frac{u^{j}}{j!} , we find :

Gen(n, m) = \frac{n!}{m!}[u^{n−m}] ∑_{i} \binom{m}{i}\frac{(−1)^{i}}{2^{i}} θ(u)^{i} = \binom{n}{m} ∑_{i=0}^{n−m} \binom{m}{i}\frac{(−1)^{i}i!}{2^{i}}θ(n − m, i)

= \binom{n}{m}(−1)^{n−m}(n − m)! ∑_{i=0}^{n−m} \binom{m}{i}\frac{i}{2^{i}} δ_{n−m−i}(n − m),

Gen(n + 1, 1) = G_{n+1} = (−1)^{n} ∑_{i=0}^{n−m} \binom{1}{i}\frac{i}{2^{i}} δ_{n−i}(n) = [n = 0] + \frac{(−1)^{n}(n + 1)!}{2}δ_{n−1}(n).

With G(u) = \frac{2u}{e^{u} + 1} = \frac{2u(e^{u} − 1)}{e^{2u} − 1} = B(2u)S(u), we get the binomial convolution
Gen(n, m) = ∑_{i=0}^{n−m} \binom{n}{i}2^{i}B_{i}^{(m)}S(n − i, m).
With \frac{Ber(2u)}{2} = u\frac{2u}{e^{2u} − 1} = \frac{u}{e^{u} − 1} \frac{2u}{e^{u} + 1} = B(u)G(u), we get :
2^{n−m}Ber(n, m) = ∑_{i=0}^{n−m} \binom{n}{i}B_{i}^{(m)}Gen(n − i, m) = \binom{n}{m} ∑_{i=0}^{n−m} \binom{n − m}{i}B_{i}^{(m)}E_{n−m−i}^{(m)}(0);

Gen(n + 1, 1) = G_{n+1} = (n + 1)E_{n}(0) = ∑_{i=0}^{n} \binom{n + 1}{i}2^{i}B_{i},

Ber(n + 1, 1) = (n + 1)B_{n} = \frac{1}{2^{n}} ∑_{i=0}^{n} \binom{n + 1}{i}B_{i}G_{n+1−i}.

Writing G(2u)/2 = 2u/(e^{2u} + 1) = e^{−u}u/ cosh u = e^{−u}Eul(u), and applying the binomial coefficients inversion formula, we obtain a pair of inverse relations, presumably original,

\frac{2^{n−m}}{m^{n}} Gen(n, m) = ∑_{i} \binom{n}{i}(−1)^{n−i} \frac{Eul(i, m)}{m^{i}} ↔ \frac{Eul(n, m)}{m^{n}} = ∑_{i} \binom{n}{i}\frac{2^{i−m}Gen(i, m)}{m^{i}},

2^{n}G_{n+1} = ∑_{i=0}^{n} \binom{n + 1}{i + 1}(−1)^{n−i}(i + 1)E_{i} ↔ (n + 1)E_{n} = ∑_{i=0}^{n} \binom{n + 1}{i + 1}2^{i}G_{i+1}.

Example 8. We consider the \underline{Harm1 -array} \{har1, Har1 = (1, Har1(u))\} and the \underline{Harm2 -array} {har2, Har2 = (1, Har2(u))}, where Har1(u) and Har2(u) are gf’s related to the harmonic numbers H_{n}:

∑_{k>0} H_{n}u^{n} = \frac{− ln(1 − u)}{1 − u} = Har1(u) [8, (7.43)],

2 ∫ \frac{− ln(1 − u)}{1 − u}du = \frac{ln^{2}(1 − u)}{u} = u (\frac{ln(1 − u)}{−u})^{2} = ∑_{n>0} \frac{2H_{n}}{n + 1}u^{n} = Har2(u),

Har1(n, m) = \frac{n!}{m!}[u^{n−m}] (\frac{− ln(1 − u)}{u(1 − u)})^{m} , Har2(n, m) = \frac{n!}{m!}[u^{n−m}] (\frac{ln(1 − u)}{−u})^{2m} .

Using the inverse pair of group representations Har1 = (−)s⋆(−)r and (−)r = (−)S ⋆Har1, and Har1(n + 1, 1)/(n + 1)! = H_{n+1}, we get:

Har1(n, m) = ∑_{i}(−1)^{n−i}s(n, i)\binom{i}{m}m^{i−m} ↔ \binom{n}{m}m^{n−m} = ∑_{i}(−1)^{n−i}S(n, i)Har1(i, m),

(n+1)!H_{n+1} = ∑_{i=0}^{n}(−1)^{n−i}s(n+1, i+1)(i+1) ↔ n+1 = ∑_{i=0}^{n}(−1)^{n−i}S(n+1, i+1)(i+1)!H_{i+1}.

Similarly, with (−)Har2 = s ⋆ Ber and Ber = S ⋆ (−)Har2, and Har2(n + 1, 1)/(n + 1)! = 2H_{n+1}/(n + 2), we find:

(−1)^{n−m}Har2(n, m) = ∑_{i=m}^{n} s(n, i)\binom{i}{m}B_{i−m}^{(m)} = \frac{n!}{(m − 1)!}2σ_{n−m}(n + m)

↔ \binom{n}{m}B_{n−m}^{(m)} = ∑_{i=m}^{n} S(n, i)(−1)^{i−m}Har2(i, m);

\frac{2H_{n+1}}{n + 2} = \frac{(−1)^{n}}{(n + 1)!} ∑_{i=0}^{n} s(n + 1, i + 1)(i + 1)B_{i} = 2σ_{n}(n + 2),  (11)

B_{n} = \frac{2}{n + 1} ∑_{i=0}^{n} S(n + 1, i + 1)\frac{(−1)^{i}(i + 1)!}{i + 2}H_{i+1}.

The identity (11) appears in [12, p.100], written in a different form. Finally:

Har1_{k}(x) = (−1)^{k} ∑_{i=0}^{k}(x − i)σ_{k−i}(k − x)(−x)^{i−1}, Har2_{k}(x) = (−1)^{k+1}2σ_{k}(k − 2x).

The elements Har1 and Har2 are related to the Narumi Polynomials Nar_{k}^{(a)}(x) [12, p.127], forming the Sheffer sequence for (g(u) = (u/(e^{u}−1))^{a}, f(u) = e^{u}−1), \overline{f}(u) = ln(1+u):

∑_{k} \frac{Nar_{k}^{(a)}(x)}{k!}u^{k} = (\frac{u}{ln(1 + u)})^{a} (1 + u)^{x}.  (12)

In fact, from (12) and the definitions of Har1 and Har2, we derive the canonical forms:

Har1(n, m) = \binom{n}{m}(−1)^{n−m}Nar_{n−m}^{(−m)}(−m), Har2(n, m) = \binom{n}{m}(−1)^{n−m}Nar_{n−m}^{(−2m)}(0).

Nar^{(a)} = U^{(a)} = ((\frac{u}{ln(1 + u)})^{a} , u(1 + u)) , C^{(a)} = ((\frac{u}{ln(1 + u)})^{a} , ln(1 + u));

Nar^{(a)}(n, m) = \binom{n}{m}Nar_{n−m}^{(a)}(m), Nar_{k}^{(a)}(x) = ∑_{i=0}^{k} C(k, i)x^{i},

C^{(a)}(n, m) = \frac{n!}{m!}[u^{n−m}] (\frac{ln(1 + u)}{u})^{m−a} = (−1)^{n−m} \frac{n!}{m!}(m − a)σ_{n−m}(n − a).

In addition, with u → −u, v = − ln(1 − u) and x = −a = m, (12) yields:

\frac{(−1)^{k}Nar_{k}^{(−m)}(m)}{k!} = [u^{k}] (\frac{ln(1 − u)(1 − u)}{−u})^{m} = [u^{k}] (\frac{v}{e^{v} − 1})^{m} = [u^{k}] ∑_{i} B_{i}^{(m)} \frac{v^{i}}{i!}

= ∑_{i=1}^{k} \frac{(−1)^{k−i}}{k!} B_{i}^{(m)}s(k, i);  (13)

when m = 1, ∑_{i=1}^{k}(−1)^{k−i}s(k, i)B_{i} = [\frac{u^{k}}{k!} ] \frac{ln(1 − u)(1 − u)}{−u} = k! (\frac{1}{k + 1} − \frac{1}{k}) = \frac{−(k − 1)!}{k + 1},

that is an identity which appears in Wilf [17, (4.3.21)].

Consider the \underline{T au-array} \{τ, T \}, τ = s ⋆ (−)r, τ(u) = −r(−s(u)) = (1 + u) ln(1 + u):

τ(n, m) = Nar^{(−m)}(n, m) = \binom{n}{m}Nar_{n−m}^{(−m)}(m) = ∑_{i=m}^{n} s(n, i)\binom{i}{i − m}m^{i−m},

T (n, m) = (−1)^{n−m}τ̃(n, m) = ∑_{i=m}^{n} S(i, m)\binom{n − 1}{n − i}n^{n−i}.

The numbers τ(n, m) are denoted b(n, m) in Comtet [4, pp. 139–140], wherein they are used in the computation of the n-th derivative of x^{ax}, x > 0, a real ≠ 0. For completeness, we also mention the group representation τ = (2)λ ⋆ Har1.

5 Concluding remarks

In this paper we obtained several known or original identities between sequences by setting m = 1 in identities between generalized numbers, for example, (11). Conversely, we found identities between generalized numbers, extending identities between numerical sequences like, for instance, (13) which is an original generalization of Wilf’s identity [17, (4.3.21)]. The Akiyama-Tanigawa algorithm and the Euler-Seidel construction are also based on extensions of numerical sequences, but the matrices have properties different from those of Riordan matrices and, at any rate, the purpose there is to develop efficient methods for calculating special sequences. The interested reader may consult the literature on the subject, for instance, [6, 11, 3, 7], in order to compare the various approaches.

Canonical forms implied Corollary 2, readily proving the existence of extensions ρ(x, x − k), that are of polynomial type when ρ has weight c = 1, and they established a connection with the family of “orthogonal” polynomials, but some care should be exercised because certain “orthogonal” polynomials may have different names, for instance, Mittag-Leffler polynomials and Meixner polynomials of the second kind M(x; 0, 0) [12, p. 126] coincide, similarly, Bernoulli polynomials of the second kind b_{k}(x) [12, p. 113] and Narumi polynomials Nar_{k}^{(−1)}(x) are the same. Recall also that, applying Proposition (3), we easily obtained the formula (5) for L_{k}^{(α)}(x) and (7) for M_{k}(x), which are found in [12, p. 109 and p. 76] after longer algebraic manipulations (and a printing error in the last expression of M_{k}(x)).

Finally, we remember that transformation rules greatly simplified algebraic manipulations, and that the duality law played an important role in the computation of inverse numbers, especially when one of the two gf’s: f(u), \overline{f}(u), was not available in closed form.

6 Acknowledgement

The author wishes to thank an anonymous referee for their careful reading of this paper and their constructive suggestions.

References

[1] A. Adelberg, Arithmetic properties of the Nörlund polynomials B_{n}^{(x)}, Discrete Math. 204 (1999), 5–13.

[2] P. Barry, Some observations on the Lah and Laguerre transforms of integer sequences, J. Integer Seq. 10 (2007), 1–18, Article 07.4.6.

[3] S. H. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Seq. 4 (2001), Article 01.1.6.

[4] L. Comtet, Advanced Combinatorics, Riedel, Dordrecht, 1974.

[5] G. Della Riccia, Converting between generalized Bell, Lah, Stirling, and Tanh numbers, J. Integer Seq. 9 (2006), Article 06.3.5.

[6] A. Dil, V. Kurt and M. Cenkci, Algorithms for Bernoulli and allied polynomials, J. Integer Seq. 10 (2007), Article 07.5.4.

[7] D. Dumont, Matrices d’Euler-Seidel, Séminaire Lotharingien de Combinatoire, 1981, Paper B=05c. (Formerly Publ. I.R.M.A. Strabourg, 1982, 182/S-04. pp. 59–78)

[8] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Sciences, Addison-Wesley, 1994.

[9] M. Kaneko, Poly-Bernoulli numbers, J. Théorie des Nombres 9 (1997), 221–228.

[10] D. E. Knuth, Convolution Polynomials, Mathematica J. 2 (4) (Fall 1992), 67–78.

[11] D. Merlini, R. Sprugnoli and M. C. Verri, The Akimaya-Tanigawa transformation, Integers: Electronic J. of Combinatorial Number Theory 5 (2005), no. 1, A05.

[12] S. Roman, The Umbral Calculus, Academic Press, 1984.

[13] L. W. Shapiro, S. Getu, W-J. Woan and L. C. Woodson, The Riordan group, Discrete Appl. Math. 34 (1991), 229–239.

[14] L. W. Shapiro, Bijections and the Riordan group, Theoret. Comp. Sci. 307 (2003), 403–413.

[15] R. Sprugnoli, Riordan arrays and combinatorial sums, Discrete Math. 132 (1994), 267– 290.

[16] E. W. Weisstein, Sheffer Sequence, From MathWorld–A Wolfram Web Resource.

    http://mathworld.wolfram.com/ShefferSequence.html .

[17] H. S. Wilf, generatingfunctionology, Academic Press, Inc., San Diego, 1990.

[18] X. Zhao and T. Wang, Some identities related to reciprocal functions, Discrete Math. 265 (2003), 323–335.

2000 Mathematics Subject Classification: Primary 05A19; Secondary 05A15. Keywords: Riordan group, convolution polynomials, polynomial extensions, Sheffer sequences, orthogonal polynomials.

Received October 17 2008; revised version received December 11 2008. Published in Journal of Integer Sequences, December 11 2008.

Return to Journal of Integer Sequences home page.