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

TL;DR

This paper introduces two-parameter generalizations of combinatorial generating functions related to ranks and Durfee symbols, revealing their automorphic properties as quasimock theta and quasimodular forms through q-series identities.

Contribution

It defines new two-parameter generalizations of Andrews' combinatorial constructs and establishes their automorphic nature as quasimock theta and quasimodular forms.

Findings

Specializations yield quasimock theta functions.

A fourth specialization produces quasimodular forms.

The automorphic properties are derived via q-series identities.

Abstract

We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews' smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using q-series identities relating the relevant generating functions to known mock theta functions.