Universal mock theta functions and two-variable Hecke-Rogers identities

TL;DR

This paper derives two-variable Hecke-Rogers identities for universal mock theta functions, revealing new double sum representations and generating function identities for various rank and spt-crank functions, using hypergeometric theory.

Contribution

It introduces new two-variable identities for universal mock theta functions and related rank functions, expanding the understanding of their structure and representations.

Findings

Many mock theta functions have Hecke-Rogers-type double sum representations.

New generating function identities for rank and spt-crank functions are established.

Results are proved using basic hypergeometric function theory.

Abstract

We obtain two-variable Hecke-Rogers identities for three universal mock theta functions. This implies that many of Ramanujan's mock theta functions, including all the third order functions, have a Hecke-Rogers-type double sum representation. We find new generating function identities for the Dyson rank function, the overpartition rank function, the M2-rank function and related spt-crank functions. Results are proved using the theory of basic hypergeometric functions.