Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols

TL;DR

This paper explores the automorphic properties of generating functions related to generalized odd rank moments and Durfee symbols, revealing their connections to modular, mock modular, and quasimodular forms.

Contribution

It introduces two-parameter generalizations of Durfee symbols and rank moments, analyzing their generating functions' automorphic properties across different parameter values.

Findings

For k=0, generating functions are modular and mock modular forms.

For k≥1, generating functions are quasimodular and quasimock modular forms.

The study extends understanding of automorphic forms associated with partition statistics.

Abstract

We define two-parameter generalizations of Andrews' $(k+1)$-marked odd Durfee symbols and $2k$th symmetrized odd rank moments, and study the automorphic properties of some of their generating functions. When $k=0$ we obtain families of modular forms and mock modular forms. When $k \geq 1$, we find quasimodular forms and quasimock modular forms.