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

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 ${}_2{\phi }_1$, and nonterminating ${}_{12}{W}_{11}$ and ${}_{10}{W}_9$.