Overpartition Rank Differences Modulo 7 By Maass Forms

TL;DR

This paper derives formulas for overpartition rank differences modulo 7 by leveraging their connection to harmonic Maass forms, improving understanding of their transformation properties and behavior at cusps.

Contribution

It provides new simple formulas for the transformation behavior of the overpartition rank function under SL(2,Z) and at cusps, enhancing the harmonic Maass form framework.

Findings

Formulas for rank differences modulo 7 are established.

Improved transformation formulas for overpartition rank functions.

Enhanced understanding of harmonic Maass forms related to overpartitions.

Abstract

Using that the overpartition rank function is the holomorphic part of a harmonic Maass form, we deduce formulas for the rank differences modulo 7. To do so we make improvements on the current state of the overpartition rank function in terms of harmonic Maass forms by giving simple formulas for the transformations under $\mbox{SL}_2(\mathbb{Z})$ as well as formulas for orders at cusps.