Generalizations of generating functions for basic hypergeometric orthogonal polynomials

TL;DR

This paper develops generalized generating functions for various basic hypergeometric orthogonal polynomials using connection relations, leading to new integral and series representations.

Contribution

It introduces a unified method to derive generalized generating functions for multiple families of basic hypergeometric orthogonal polynomials.

Findings

Derived generalized generating functions for Askey-Wilson and other polynomials.

Established new integral and series formulas related to these polynomials.

Extended the theory of generating functions in the context of basic hypergeometric orthogonal polynomials.

Abstract

We derive generalized generating functions for basic hypergeometric orthogonal polynomials by applying connection relations with one free parameter to them. In particular, we generalize generating functions for the Askey-Wilson, continuous $q$-ultraspherical/Rogers, little $q$-Laguerre/Wall, and $q$-Laguerre polynomials. Depending on what type of orthogonality these polynomials satisfy, we derive corresponding definite integrals, infinite series, bilateral infinite series, and $q$-integrals.