Utility of integral representations for basic hypergeometric functions and orthogonal polynomials

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,...

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Bibliographic Details
Published in:The Ramanujan journal Vol. 61; no. 2; pp. 649 - 674
Main Authors: Cohl, Howard S., Costas-Santos, Roberto S.
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2023
Subjects:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
Basic hypergeometric functions
Integral representations
Basic hypergeometric orthogonal polynomials
Generating functions
33D60
Transformations
33D15
ISSN:1382-4090, 1572-9303
Online Access:Get full text
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Summary: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.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-021-00509-5