# On a generalization of the Rogers generating function

Research paper by Howard S. Cohl, Roberto S. Costas-Santos, Tanay Wakhare

Indexed on: 24 May '18Published on: 24 May '18Published in: arXiv - Mathematics - Classical Analysis and ODEs

#### 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 ${}_8\phi_7$.