The generating function of the $M_2$-rank of partitions without repeated odd parts as a mock modular form

TL;DR

This paper advances the understanding of the $M_2$-rank generating function for partitions without repeated odd parts by establishing its harmonic Maass form properties and deriving formulas for rank differences modulo various integers.

Contribution

It improves the harmonic Maass form status of the generating function and derives new formulas for rank differences modulo any integer greater than 2.

Findings

The generating function transforms like a modular form.

Formulas for rank differences modulo 7 are obtained.

General identities for rank differences modulo any c > 2 are provided.

Abstract

By work of Bringmann, Ono, and Rhoades it is known that the generating function of the $M_2$-rank of partitions without repeated odd parts is the so-called holomorphic part of a certain harmonic Maass form. Here we improve the standing of this function as a harmonic Maass form and show more can be done with this function. In particular we show the related harmonic Maass form transforms like the generating function for partitions without repeated odd parts (which is a modular form). We then use these improvements to determine formulas for the rank differences modulo $7$. Additionally we give identities and formulas that allow one to determine formulas for the rank differences modulo $c$, for any $c>2$.