Nonterminating transformations and summations associated with some q-Mellin–Barnes integrals - Cohl, Howard S.; Costas-Santos, Roberto S. (Advances in applied mathematics, 2023)
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 ϕ23, nonterminating very-well-poised W45, W78, products of two nonterminating ϕ12's, square of a nonterminating well-poised ϕ12, a nonterminating W910, two nonterminating W1112's and several nonterminating summations which arise from the Askey–Roy and Gasper integrals.