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

TL;DR

This paper derives new q-Mellin--Barnes integrals and uses them to establish transformation and summation formulas for various nonterminating basic hypergeometric functions, expanding the analytical tools available for these integrals.

Contribution

The paper introduces novel q-Mellin--Barnes integrals and applies them to derive transformation and summation formulas for complex nonterminating hypergeometric functions.

Findings

Derived new q-Mellin--Barnes integrals

Established transformation formulas for nonterminating hypergeometric functions

Provided summation formulas for various complex hypergeometric series

Abstract

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, square of a nonterminating well-poised ${}_2\phi_1$, and nonterminating ${}_{12}W_{11}$ and ${}_{10}W_9$.