Utility of integral representations for basic hypergeometric functions and orthogonal polynomials

TL;DR

This paper explores integral representations of basic hypergeometric functions and orthogonal polynomials, deriving transformations, generalizations, and applications to generating functions within the $q$-Askey scheme.

Contribution

It introduces new transformation formulas and generalizations for integrals related to basic hypergeometric functions and orthogonal polynomials, expanding their theoretical framework.

Findings

Derived an infinite sequence of symmetrization transformations.

Expressed certain integrals as symmetric sums of basic hypergeometric series.

Applied integral representations to generate functions for Askey--Wilson and continuous dual q-Hahn polynomials.

Abstract

We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integrals connected to nonterminating basic hypergeometric series and the Askey--Wilson polynomials. We express a certain integral of a ratio of infinite $q$-shifted factorials as a symmetric sum of two basic hypergeometric series with argument $q$. The result is then expressed as a $q$-integral. Examples of integral representations applied to the derivation of generating functions for Askey--Wilson are given and as well the computation of a missing generating function for the continuous dual $q$-Hahn polynomials.