On three third order mock theta functions and Hecke-type double sums

TL;DR

This paper derives new Hecke-type double sums for Ramanujan's third order mock theta functions, explores their connections to recent developments in q-orthogonal polynomials and topological invariants, and proves identities among these functions.

Contribution

It introduces four new Hecke-type double sums for three third order mock theta functions and links them to recent advances in mock theta functions and topological invariants.

Findings

Derived four Hecke-type double sums for mock theta functions

Connected these sums to q-orthogonal polynomials and Witten-Reshetikhin-Turaev invariants

Proved identities expressing mock theta functions in terms of the universal mock theta function

Abstract

We obtain four Hecke-type double sums for three of Ramanujan's third order mock theta functions. We discuss how these four are related to the new mock theta functions of Andrews' work on $q$-orthogonal polynomials and Bringmann, Hikami, and Lovejoy's work on unified Witten-Reshetikhin-Turaev invariants of certain Seifert manifolds. We then prove identities between these new mock theta functions by first expressing them in terms of the universal mock theta function.