Some multilinear generating functions

Abstract

The multiple hypergeometric generating function (1.3) below, due to H. M. Srivastava and J. P. Singhal [Acad. Roy. Belg. Bull. Cl. Sci. (5)58 (1972), 1238–1247], applies readily to deduce multilinear generating functions for thespecial Jacobi polynomialsP ^(α-n,β)_n (x), P ^(α,β-n)_n (x) orP ^(α-n,β-n)_n (x), the Laguerre polynomialsL ^(α)_n (x), thebiorthogonal polynomialsZ ^α_n (x; k) of J.D.E. Konhauser [Pacific J. Math. 21 (1967), 303–314], and so on, and indeed also for any suitable products of these polynomials. The present paper is motivated by the need for a multiple hypergeometric generating function, analogous to (1.3), which could apply to yield multilinear generating functions for theunrestricted Jacobi polynomialsP ^(α,β)_n (x). Several interesting generalizations of the multiple hypergeometric generating function (1.3), and of its analogue (5.6) thus obtained, are given; many of these generalizations are shown to apply also to derive multilinear generating functions for the classical Hermite polynomialsH _n(x) and for their various known generalizations considered, among others, by F. Brafman [Canad. J. Math. 9 (1957), 180–187] and by H. W. Gould and A. T. Hopper [Duke Math. J. 29 (1962), 51–63].

The multilinear generating functions (1.19), (1.22), (1.23), (1.25), (1.30), (3.3), (4.1), (4.2), (4.8), (5.5), (5.6), (6.3), (6.4) and (6.6) below are believed to be new.