Utility of integral representations for basic hypergeometric functions and orthogonal polynomials

Published

June 23, 2022

Author(s)

Roberto S. Costas-Santos, Howard Cohl

Abstract

We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the q-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integrals connected to nonterminating basic hypergeometric series and the Askey–Wilson polynomials. We express a certain integral of a ratio of infinite q-shifted factorials as a symmetric sum of two basic hypergeometric series with argument q. The result is then expressed as a q-integral. Examples of integral representations applied to the derivation of generating functions for the Askey–Wilson polynomials are given and as well the computation of a missing generating function for the continuous dual q-Hahn polynomials.

Citation

The Ramanujan Journal

Volume

61

Pub Type

Journals

Download Paper

https://doi.org/10.1007/s11139-021-00509-5

Local Download

Keywords

Basic hypergeometric functions · Transformations · Integral representations · Basic hypergeometric orthogonal polynomials · Generating functions

Citation

Costas-Santos, R. and Cohl, H. (2022), Utility of integral representations for basic hypergeometric functions and orthogonal polynomials, The Ramanujan Journal, [online], https://doi.org/10.1007/s11139-021-00509-5, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=932468 (Accessed October 13, 2025)