On a generalization of the Rogers generating function

Published

July 14, 2019

Author(s)

Howard Cohl, Roberto Costas-Santos, Tanay Wakhare

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.

Citation

Journal of Mathematical Analysis and Applications

Volume

475

Issue

2

Pub Type

Journals

Download Paper

https://doi.org/10.1016/j.jmaa.2019.01.068

Local Download

Keywords

Basic hypergeometric series, Basic hypergeometricorthogonal polynomials, Generating functions, Connectioncoefficients, Eigenfunction expansions, Definite integrals.

Citation

Cohl, H. , Costas-Santos, R. and Wakhare, T. (2019), On a generalization of the Rogers generating function, Journal of Mathematical Analysis and Applications, [online], https://doi.org/10.1016/j.jmaa.2019.01.068, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=925841 (Accessed October 14, 2025)