Rank and crank moments for overpartitions

TL;DR

This paper investigates crank and rank moments for overpartitions, expressing them via quasimodular forms, and derives exact relations and congruences, including connections to Hurwitz class numbers.

Contribution

It introduces new relations between overpartition moments and quasimodular forms, and establishes novel congruences and connections to class numbers.

Findings

Crank moments and derivatives can be expressed as quasimodular forms.

Exact relations and congruences modulo 3, 5, and 7 are proved.

A new congruence involving overpartition functions and Hurwitz class numbers is established.

Abstract

We study two types of crank moments and two types of rank moments for overpartitions. We show that the crank moments and their derivatives, along with certain linear combinations of the rank moments and their derivatives, can be written in terms of quasimodular forms. We then use this fact to prove exact relations involving the moments as well as congruence properties modulo 3, 5, and 7 for some combinatorial functions which may be expressed in terms of the second moments. Finally, we establish a congruence modulo 3 involving one such combinatorial function and the Hurwitz class number H(n).