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arXiv · 1805.10149

On a generalization of the Rogers generating function

Howard S. Cohl·Roberto S. Costas-Santos·Tanay Wakhare
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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 ${}_8\phi_7$.

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Howard S. Cohl, Roberto S. Costas-Santos, Tanay Wakhare. 2018-05-24. On a generalization of the Rogers generating function. https://arxiv.org/abs/1805.10149

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