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Nonterminating transformations and summations associated with some q-Mellin-Barnes integrals



Howard Cohl, Roberto Costas-Santos


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 $}_5W_4$, $}_8W_7$, products of two nonterminating $}_2\phi_1$'s and squares of a nonterminating well-poised $}_2\phi_1$.
Advances in Applied Mathematics


q-calculus, Nonterminating basic hypergeometric functions, Nonterminating transformations, Nonterminating summations, Integral representations, q-Mellin–Barnes integrals, Askey–Wilson polynomials, Askey–Wilson moments


Cohl, H. and Costas-Santos, R. (2023), Nonterminating transformations and summations associated with some q-Mellin-Barnes integrals, Advances in Applied Mathematics, [online],, (Accessed April 19, 2024)
Created March 1, 2023, Updated May 3, 2023