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On a generalization of the Rogers generating function

Published

Author(s)

Howard S. 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
Constructive Approximation
Volume
475
Issue
2

Keywords

Basic hypergeometric series, Basic hypergeometric orthogonal polynomials, Generating functions, Connection coefficients, Eigenfunction expansions, Definite integrals.
Created July 14, 2019, Updated October 8, 2020