Integral representations for the Lagrange polynomials, Shively’s pseudo-Laguerre polynomials, and the generalized Bessel polynomials

Abstract

Motivated essentially by their potential for applications in the mathematical, physical, and statistical sciences, the object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing the main results presented here, the corresponding integral representations are derived for familiar simpler classes of hypergeometric polynomials such as (for example) the Lagrange polynomials, Shively's pseudo-Laguerre polynomials, and generalized Bessel polynomials. Each of the integral representations derived in this paper may be also viewed as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.

Some generating functions of the Bessel and related orthogonal polynomials

“…Various further developments emerging from Burchnall's work [4] can be found in the recent surveycum-expository review article by Srivastava [37] (see also the references which are cited therein). In this connection, we record here the following development which emerged recently from Burchnall's work [4] (see, for details, [37, Eq.…”

“…For detailed descriptions of the success and usefulness of several families of hypergeometric generating functions in the derivation of simpler generating functions for numerous classes of hypergeometric polynomials, including (for example) the simple Bessel polynomials y n (x) and the generalized Bessel polynomials Y (α,β) n (x), can be found in the earlier works [36], [37], [38] and [39]. For example, the following general families of generating functions involving an appropriately bounded sequence {Ω(n)} n∈N 0 of essentially arbitrary real or complex numbers (see, for details, [36]):…”

Linearization relations for the generalized Bedient polynomials of the first and second kinds via their integral representations

“…], and Lin et al [6], [9, p. 448 et seq. ], [8] and [7], Liu et al [10] and Srivastava et al [14] and [16]; see also [5]). Here, in our present investigation, we consider the polynomial family defined by where (and throughout this paper) (L s ) abbreviates the array of s parameters…”