An infinite family of vector-valued mock theta functions

TL;DR

This paper constructs an infinite family of vector-valued mock theta functions linked to harmonic Maass forms, extending previous results and deepening understanding of their modular transformation properties.

Contribution

It introduces a new infinite family of mock theta functions derived from Dyson's rank generating function and related functions, with associated harmonic Maass forms transforming under the Weil representation.

Findings

Strengthens previous results by Bringmann and Ono (2010)

Extends Garvan's (2019) work on mock theta functions

Provides explicit construction of vector-valued mock theta functions

Abstract

We exhibit an infinite family of vector-valued mock theta functions indexed by positive integers coprime to $6$. These are built from specializations of Dyson's rank generating function and related functions studied by Watson, Gordon, and McIntosh. The associated completed harmonic Maass forms transform according to the Weil representation attached to a rank one lattice. This strengthens a 2010 result of Bringmann and Ono and a 2019 result of Garvan.