Quasimodularity and Limiting Behavior for Variations of MacMahon Series

TL;DR

This paper extends the family of $q$-series related to MacMahon's work, showing they are quasimodular forms with determined weights and levels, and explores their limiting behavior and connections to partition functions.

Contribution

It broadens the class of $q$-series known to be quasimodular, precisely characterizes their weights and levels, and analyzes their asymptotic and reciprocal properties.

Findings

The extended family of $q$-series are quasimodular forms.

The weights and levels of these forms are explicitly determined.

The sequence $\

Abstract

Motivated by the 1920's seminal work of Major MacMahon, Amdeberhan--Andrews--Tauraso recently introduced an infinite family of $q$-series \[ \mathcal{U}_{t}(a;q):= \sum_{1\le n_1<n_2<\cdots<n_t} \frac{q^{n_1+n_2+\cdots+n_t}}{(1+aq^{n_1}+q^{2n_1})(1+aq^{n_2}+q^{2n_2})\cdots (1+aq^{n_t}+q^{2n_t})} \] and proved that these functions are linear combinations of quasimodular forms. In this paper, we study a broader family of $q$-series that contains the collection $\{\mathcal{U}_t\}_{t \in \mathbb{N}}$. Using the theory of quasi shuffle algebras, we show that this extended family also lies in the algebra of quasimodular forms. Moreover, we determine the precise weights and levels of these functions, thereby making Amdeberhan--Andrews--Tauraso's result sharp. We further investigate the limiting behavior of these functions. In particular, we demonstrate that the sequence of quasimodular forms~$\{\mathcal{U}_t(1;q)\}_{t\in\mathbb{N}}$ gives an approximation for the ordinary partition function. We also establish infinitely many closed formulas for reciprocals of certain infinite products in terms of~$\mathcal{U}_{t}(a;q)$.