Mock theta double sums

TL;DR

This paper develops a general framework for Bailey pairs, introduces new Bailey pairs, and constructs novel q-hypergeometric double sums that are identified as mock theta functions, connecting them to classical mock theta functions.

Contribution

It generalizes Bailey pair theory, derives new Bailey pairs, and constructs new mock theta double sums with identities linking them to classical functions.

Findings

New Bailey pairs of a similar type to existing ones.

Construction of new q-hypergeometric double sums as mock theta functions.

Identities between new double sums and classical mock theta functions.

Abstract

We prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.