On a generalization of the Rogers generating function

TL;DR

This paper extends the Rogers generating function to continuous q-ultraspherical polynomials, deriving new integral formulas and hypergeometric transformations by leveraging connection relations and recent generalizations.

Contribution

It introduces a generalized Rogers generating function for continuous q-ultraspherical polynomials and derives new hypergeometric transformations using recent Askey-Wilson polynomial expansions.

Findings

Derived a generalized Rogers generating function for continuous q-ultraspherical polynomials.

Established a new quadratic transformation connecting ${}_2 ext{phi}_1$ and ${}_8 ext{phi}_7$ series.

Extended the expansion framework to continuous q-Jacobi and Wilson polynomials.

Abstract

We derive a generalized Rogers generating function and corresponding definite integral, for the continuous $q$-ultraspherical polynomials by applying its connection relation and utilizing orthogonality. Using a recent generalization of the Rogers generating function by Ismail & Simeonov expanded in terms of Askey-Wilson polynomials, we derive corresponding generalized expansions for the continuous $q$-Jacobi, and Wilson polynomials with two and four free parameters respectively. Comparing the coefficients of the Askey-Wilson expansion to our continuous $q$-ultraspherical/Rogers expansion, we derive a new quadratic transformation for basic hypergeometric series connecting ${}_2\phi_1$ and ${}_8\phi_7$.