A Generalization of MacMahon Series via Cyclotomic Polynomials

TL;DR

This paper introduces a broad generalization of MacMahon series using cyclotomic polynomials, revealing their quasimodular nature and expressing them as isobaric polynomials, thus extending prior work and connecting to existing generalizations.

Contribution

It defines new series based on cyclotomic polynomials, proves their quasimodularity, and relates them to existing generalizations, expanding the theoretical framework of MacMahon series.

Findings

Series are quasimodular forms of higher weight and level

Explicit representations involve Eulerian polynomials and Gauss sums

Special cases recover known generalizations

Abstract

About a century ago, P. A. MacMahon introduced a class of $q$-series, which are nowadays referred to as MacMahon series. More recently, in 2013, G. E. Andrews and S. C. F. Rose revealed the quasimodular property of these series. In this paper, we introduce a generalization of MacMahon series. Specifically, for any positive integers $t, k, N$ and a polynomial $Q(x)$, we define the series $\mathcal{U}_{t, k; N}(Q; q)$ and $\mathcal{U}_{t, k; N}^{\star}(Q; q)$ using the $N$-th cyclotomic polynomial. To investigate these series, we apply a decomposition formula involving the Eulerian polynomials and express the $N$-th roots of unity in terms of Gauss sums. By combining these results to derive explicit representations, we prove that our series arise as quasimodular forms of higher weight and higher level. Furthermore, we show that they can be expressed as isobaric polynomials. In particular, we show that the one-parameter generalization introduced by C. Nazaroglu, B. V. Pandey, and A. Singh arises as a special case of our theory.