On the Askey–Wilson polynomials and a 𝑞-beta integral

Liu, Zhi-Guo

doi:10.1090/proc/15584

On the Askey–Wilson polynomials and a $q$-beta integral
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by Zhi-Guo Liu

Proc. Amer. Math. Soc. 149 (2021), 4639-4648

DOI: https://doi.org/10.1090/proc/15584

Published electronically: August 5, 2021

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Abstract:

A proof of the orthogonality relation for the Askey–Wilson polynomials is given by using a generating function for the Askey–Wilson polynomials and the uniqueness of a rational function expansion. We further use the orthogonality relation for the Askey–Wilson polynomials and a $q$-series transformation formula to evaluate a general $q$-beta integral with eight parameters. The integrand of this $q$-beta integral is the product of two terminating $_5\phi _4$ series and the value is a $_{10}\phi _9$ series.

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