Some Families of Generating Functions Associated with the Stirling Numbers of the Second Kind

Abstract

DEDICATED TO PROFESSOR GEORGE LEITMANNThe object of this paper is to present a systematic introduction to (and several interesting applications of) a general result on generating functions (associated with the Stirling numbers of the second kind) for a fairly wide variety of special functions and polynomials in one, two, and more variables. The main results given below are shown to apply not only to the classical orthogonal polynomials including, for example, the Jacobi polynomials (which contain, as their special cases, the Gegenbauer or ultraspherical polynomials, the Legendre or spherical polynomials, and the Chebyshev polynomials of the first and second kinds) and the Laguerre polynomials, and to their various extensions and generalizations studied in recent years, but indeed also to a class of generalized hypergeometric functions, the Lauricella polynomials in several variables, and the familiar Lagrange polynomials which arise in certain problems in statistics. Relevant connections of some of these families of generating functions with various known results are also indicated.

A new result for hypergeometric polynomials

“…Next we recall (as Theorem 1 below) some general results of Srivastava [2] on generating functions associated with the Stirling numbers S (n, k) of the second kind, defined by…”

“…As already remarked by Bavinck [1, p. L279], the relationship (1) was encountered in connection with certain differential operators for generalized Laguerre polynomials. The main object of this sequel to Bavinck's work [1] is to present an interesting generalization of the summation formula (1) to hold true for the classical hypergeometric polynomials 2 …”

New Generating Functions for a Class of Generalized Hermite Polynomials

“…Srivastava [6] applied Theorem 1 (as well as its multivariable extension also given by him [6, p. 765, Theorem 2]) in order to derive generating functions of the class (1.7) for a remarkably large number of special functions and polynomials in one, two, and more variables. In this sequel to the work of Srivastava [6], we aim at presenting analogous and other families of linear, bilinear, and mixed multilateral generating functions for a certain generalization of the classical Hermite (and classical Laguerre) polynomials, defined by Gould and Hopper [2] …”

“…where the coefficients A n k are constants, real or complex, and f g h are suitable functions of x and t. Recently, Srivastava [6] investigated a widely applicable special case of the Singhal-Srivastava generating function (1.1) when…”

On a sequence of functions Vn (α,β,δ) (x;a, k, s)

“…The following theorem is useful to get some generating relation for V (α,β,δ) n (x; a, k, s); Theorem 4.1 (Srivastava [10]). Let the sequence {ζ n (x)} ∞ n=0 be generated by…”