Title Utility of integral representations for basic hypergeometric functions and orthogonal polynomials
Authors Cohl, Howard S. , COSTAS SANTOS, ROBERTO SANTIAGO
External publication Si
Means Ramanujan J.
Scope Article
Nature Científica
JCR Quartile 3
SJR Quartile 2
JCR Impact 0.70000
SJR Impact 0.55900
Web https://www.scopus.com/inward/record.uri?eid=2-s2.0-85132548888&doi=10.1007%2fs11139-021-00509-5&partnerID=40&md5=8fce344953c9af4a43568a803850a122
Publication date 23/06/2022
ISI 000814913900001
Scopus Id 2-s2.0-85132548888
DOI 10.1007/s11139-021-00509-5
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.
Keywords Basic hypergeometric functions; Transformations; Integral representations; Basic hypergeometric orthogonal polynomials; Generating functions
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