A note on generating functions and summation formulae for Meixner polynomials of several variables

Abstract The present paper deals with certain generating functions and various elegant summation formulae for Meixner polynomials of several variables.


Introduction
Generalized functions occupy the place of pride in literature on special functions. Their importance which is mounting everyday stems from the fact that they generalize well-known one variable special functions namely Hermite polynomials, Laguerre polynomials, Legendre polynomials, Gegenbauer polynomials, Jacobi polynomials, Rice polynomials, generalized Sylvester polynomials, Meixner polynomials etc. All these polynomials are closely associated with problems of applied nature. For example, Gegenbauer polynomials are deeply connected with axially symmetric potentials in n dimensions and contain the Legendre and Chebyshev polynomials as special cases. The hypergeometric functions of which the Jacobi polynomials are a special case, are important in many cases of mathematical analysis and its applications. Further, Bessel functions are closely associated with problems possessing circular or cylindrical symmetry. For example, they arise in the study of free vibrations of a circular membrane and in finding the temperature distribution in a circular cylinder. They also occur in electromagnetic theory and numerous other areas of physics and engineering.
The following results are required in this paper

Definitions:
Following the work of Riordan [5] (p. 90 et seq.), one denotes by S(n, k) the Stirling numbers of the second kind, defined by and S(n, 1) = S(n, n) = 1 and S(n, n − 1) = n 2 .
Recently, several authors (see, for example, Gabutti and Lyness [2], Mathis and Sismondi [8], and Srivastava [3]) considered various families of generating functions associated with the Stirling numbers S(n, k). We choose to recall here the following general results on these families of generating functions, which were given by Srivastava [3].
where f , g and h are suitable functions of x and t. Then, in terms of the Stirling numbers S(n, k) defined by (1.1), the following family of generating functions holds true: provided that each member of (1.3) exists.
where a and b are complex constants such that |a| + |b| > 0. Also let {λ n } be any sequence of complex numbers for which If we differentiate both sides of (1.10) with respect to w, using the relationship (1.9), and replace f ′ (z)φ(z) in the resulting equation by f (z), we can write (1.10) in the form [cf. Polya and Szego (1972), p. 146, problem 207]: which is usually more suitable to apply than (1.10). For φ(z) ≡ 1, both (1.10) and (1.11) evidently yield Taylor,s expansion where, as usual, [6], p. 521) Let A(z), B(z) and z −1 C(z) be arbitrary functions which are analytic in the neighborhood of the origin, and assume that

Define the sequence of functions {f
where α and x are arbitrary complex numbers independent of z. Then, for arbitrary parameters λ and y independent of z,

Meixner polynomials
The Meixner polynomials are denoted by m n (x; β, c) and are defined as (see [10]) where β > 0, 0 < c < 1 and x = 0, 1, 2, . . . Agarwal and Manocha [1] defined the polynomials m n (x; β, c) by the generating relations The following generating function is well-known [4], p. 443, problem 5(ii) Generating functions and summation formulae 55 By applying Theorem 1, one immediately obtain the following (presumably new) generating function for the Meixner polynomials defined by (2.1): where S(n, k) and A n are given by (1.1) and (1.7) of this paper.

Applications of Carlitz's theorem
Let γ and δ be arbitrary constants. Then the polynomials m n (x; β, c) defined by (2.2) above satisfy the following generating relations: where u is a function of t defined by Proof of (3.1). We know generating function Expanding the function on the R.H.S. of (i) by Taylor's theorem.
Replacing β by β + γn in (ii) we get We know that the Lagrange's expansion formula: Therefore we get (3.1) Proof of (3.2). We know generating function Expanding the function on the R.H.S. of (i) by Taylor's theorem We know that the Lagrange's expansion formula: . Here Therefore we get (3.2) and , where w is a function of t defined by Proof of (3.3). We know generating function Expanding the function on the R.H.S. of (i) by Taylor's theorem Replacing x by x + γn and β by β + δn in (ii) we get We know that the Lagrange's expansion formula: Therefore we get (3.3).

Summation formulae for Meixner polynomials
The following summation formulae are easily derivable from known results in view of the relationship (2.2):  Proof of (4.1).

Summation formulae for Meixner polynomials of two variables
The following summation formulae holds for (5.2): Proof of (6.2).

Meixner polynomials of three variables
The Meixner polynomials of three variables m n (x, y, z; β, c, d, e, ) are defined in terms of Kampe de Feriet triple hypergeometric functions as (see [4]) follows: