A new identity for the sum of products of generalized basic hypergeometric functions

TL;DR

This paper establishes a duality relation for generalized basic hypergeometric functions, extending previous identities and deriving new multi-term relations, with implications for both terminating and non-terminating series.

Contribution

It introduces a novel duality relation for generalized basic hypergeometric functions, generalizing prior identities and deriving new multi-term relations.

Findings

Derived a duality relation extending existing hypergeometric identities.

Obtained new multi-term relations for generalized basic hypergeometric series.

Provided explicit examples and confluent versions of the main results.

Abstract

We prove a duality relation for generalized basic hypergeometric functions. It forms a $q$-extension of a recent result of the second and the third named authors and generalizes both a $q$-hypergeometric identity due to the third named author (jointly with Feng and Yang) and a recent identity for the Heine's ${}_2\phi_{1}$ function due to Suzuki. We further explore various consequences of our identity leading to several presumably new multi-term relations for both terminating and non-terminating generalized basic hypergeometric series. Moreover, we give confluent versions of our results and furnish a number of explicit examples.