# Quantum modular forms and singular combinatorial series with repeated roots of unity

## TL;DR

This paper demonstrates that the combinatorial generating functions related to Durfee symbols exhibit quantum modular properties, linking combinatorics, number theory, and quantum modular forms.

## Contribution

It establishes the quantum modularity of the functions $R_n$, generalizing previous work on their automorphic properties and connecting combinatorial series to quantum modular forms.

## Key findings

- Proves quantum modular properties of $R_n$ functions.
- Links combinatorial generating functions to quantum modular forms.
- Extends understanding of automorphic properties in partition theory.

## Abstract

In 2007, G.E. Andrews introduced the $(n+1)$-variable combinatorial generating function $R_n(x_1,x_2,\cdots,x_n;q)$ for ranks of $n$-marked Durfee symbols, an $(n+1)$-dimensional multisum, as a vast generalization to the ordinary two-variable partition rank generating function. Since then, it has been a problem of interest to understand the automorphic properties of this function; in special cases and under suitable specializations of parameters, $R_n$ has been shown to possess modular, quasimodular, and mock modular properties when viewed as a function on the upper half complex plane $\mathbb H$, in work of Bringmann, Folsom, Garvan, Kimport, Mahlburg, and Ono. Quantum modular forms, defined by Zagier in 2010, are similar to modular or mock modular forms but are defined on the rationals $\mathbb Q$ as opposed to $\mathbb H$, and exhibit modular transformations there up to suitably analytic error functions in $\mathbb R$; in general, they have been related to diverse areas including number theory, topology, and representation theory. Here, we establish quantum modular properties of $R_n$.