The \(M_2\)-rank of partitions without repeated odd parts as a harmonic Maass form

Abstract

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 \(c>2\).