The utility of integral representations for the Askey-Wilson polynomials and their symmetric sub-families

Abstract: The Askey-Wilson polynomials are a class of orthogonal polynomials which are symmetric in four free parameters which lie at the very top of the q-Askey scheme of basic hypergeometric orthogonal polynomials. These polynomials, and the polynomials in their subfamilies, are usually defined in terms of their finite series representations which are given in terms of terminating basic hypergeometric series. However, they also have nonterminating, q-integral, and integral representations. In this talk, we will explore some of what is known about the symmetry of these representations and how they have been used to compute their important properties such as generating functions. This study led to an extension of interesting contour integral representations of sums of nonterminating basic hypergeometric functions initially studied by Bailey, Slater, Askey, Roy, Gasper and Rahman. We will also discuss how these contour integrals are deeply connected to the properties of the symmetric basic hypergeometric orthogonal polynomials.

classical analysis and ODEscombinatoricsnumber theory

Audience: researchers in the topic

Special Functions and Number Theory seminar

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