The q-WZ Method for Infinite Series

TL;DR

This paper extends the q-WZ method to handle infinite series by incorporating infinite q-shifted factorials, enabling proofs of classical basic hypergeometric identities.

Contribution

It introduces a novel extension of the q-WZ method to infinite series using q-shifted factorials, broadening its applicability.

Findings

Derived q-WZ pairs for classical identities

Extended the q-WZ method to infinite series

Provided proofs for q-Gauss, $_6\phi_5$, $_1\psi_1$, and $_6\psi_6$ sums

Abstract

Motivated by the telescoping proofs of two identities of Andrews and Warnaar, we find that infinite q-shifted factorials can be incorporated into the implementation of the q-Zeilberger algorithm in the approach of Chen, Hou and Mu to prove nonterminating basic hypergeometric series identities. This observation enables us to extend the q-WZ method to identities on infinite series. As examples, we will give the q-WZ pairs for some classical identities such as the q-Gauss sum, the $_6\phi_5$ sum, Ramanujan's $_1\psi_1$ sum and Bailey's $_6\psi_6$ sum.