The $M_2$-rank of partitions without repeated odd parts as a harmonic Maass form

Chris Jennings-Shaffer

Published 2016-01-25Version 1

While it is known that the $M_2$-rank of partitions without repeated odd parts is the so-called holomorphic part of a certain harmonic Maass form, much more can been done with this fact. We greatly improve the standing of this function as a harmonic Maass form, 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 $c>2$.