Some Generating Functions for the Generalized Bessel Polynomials

Abstract

The authors begin by presenting a systematic (historical) account of some linear generating functions for the generalized Bessel polynomials. It is then shown how these linear generating functions can be applied with a view to obtaining various new families of bilinear, bilateral, or mixed multilateral generating functions for the generalized Bessel polynomials. Some interesting generalizations of these classes of generating functions, involving hypergeometric functions, are also considered.

Some generating functions of the Bessel and related orthogonal polynomials

“…In light of the relationships (2.4), (2.5) and (2.6), generating functions for the Jacobi and Laguerre polynomials can be applied also to derive the corresponding generating functions for the generalized Bessel polynomials Y (α,β) n (x). We list below some of such consequences from the generating functions for the Jacobi, Laguerre and other hypergeometric polynomials (see, for details, [8], [9], [24], [27], [32], [35], [40], [41,Chapter 7] and [42]). We choose to present here the following generating function which is derivable by appealing appropriately to Gould's combinatorial identity (3.6).…”