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Cohl, Howard S.; Costas-Santos, Roberto S.

Nonterminating transformations and summations associated with some \(q\)-Mellin-Barnes integrals. (English) Zbl 07680278

Adv. Appl. Math. 147, Article ID 102517, 46 p. (2023).
Summary: In many cases one may encounter an integral which is of \(q\)-Mellin-Barnes type. These integrals are easily evaluated using theorems which have a long history dating back to Slater, Askey, Gasper, Rahman and others. We derive some interesting \(q\)-Mellin-Barnes integrals and using them we derive transformation and summation formulas for nonterminating basic hypergeometric functions. The cases which we treat include ratios of theta functions, the Askey-Wilson moments, nonterminating well-poised \({}_3 \phi_2\), nonterminating very-well-poised \({}_5 W_4, {}_8 W_7\), products of two nonterminating \({}_2 \phi_1\)’s, square of a nonterminating well-poised \({}_2 \phi_1\), a nonterminating \({}_{10}W_9\), two nonterminating \({}_{12}W_{11}\)’s and several nonterminating summations which arise from the Askey-Roy and Gasper integrals.

MSC:

33D60 Basic hypergeometric integrals and functions defined by them
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
05A30 \(q\)-calculus and related topics

Keywords:

\(q\)-calculus; nonterminating basic hypergeometric functions; nonterminating transformations; nonterminating summations; integral representations; \(q\)-Mellin-Barnes integrals; Askey-Wilson polynomials; Askey-Wilson moments
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\textit{H. S. Cohl} and \textit{R. S. Costas-Santos}, Adv. Appl. Math. 147, Article ID 102517, 46 p. (2023; Zbl 07680278)
Full Text: DOI arXiv

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