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A Generalization of MacMahon Series via Cyclotomic Polynomials

Riku Shintani

TL;DR

The paper generalizes MacMahon’s $q$-series by introducing $\mathcal{U}_{t,k;N}(Q;q)$ and its star variant, defined with $N$-th cyclotomic denominators. It proves these series are mixed-weight quasimodular forms of weight at most $tk$ and level at most $N$, and that they admit an isobaric polynomial expression in terms of base series $\mathcal{U}_{sk;N}(Q^s;q)$. The analysis hinges on a blend of Eulerian-polynomial decompositions, Gauss-sum expansions, and Dirichlet-character techniques, with explicit $N=1,2$ and general $N\ge3$ cases. The Nazaroglu–Pandey–Singh one-parameter generalization emerges as a special instance of the established framework, highlighting the broad reach of the cyclotomic MacMahon generalization and its quasimodular structure.

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.

A Generalization of MacMahon Series via Cyclotomic Polynomials

TL;DR

The paper generalizes MacMahon’s -series by introducing and its star variant, defined with -th cyclotomic denominators. It proves these series are mixed-weight quasimodular forms of weight at most and level at most , and that they admit an isobaric polynomial expression in terms of base series . The analysis hinges on a blend of Eulerian-polynomial decompositions, Gauss-sum expansions, and Dirichlet-character techniques, with explicit and general cases. The Nazaroglu–Pandey–Singh one-parameter generalization emerges as a special instance of the established framework, highlighting the broad reach of the cyclotomic MacMahon generalization and its quasimodular structure.

Abstract

About a century ago, P. A. MacMahon introduced a class of -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 and a polynomial , we define the series and using the -th cyclotomic polynomial. To investigate these series, we apply a decomposition formula involving the Eulerian polynomials and express the -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.

Paper Structure

This paper contains 11 sections, 13 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eisenstein Series and Quasimodular Forms
  4. Eulerian Polynomials
  5. Gauss Sums
  6. Proof of Theorem \ref{['main_thm']} \ref{['quasimodularity_main']}
  7. The Case N = 1
  8. The Case N = 2
  9. The Case N ≥ 3
  10. Proof of Theorem \ref{['main_thm']} \ref{['decomposition_main']}
  11. Explicit formulas for a_{N, j; Q}(r) and c_{N, j; Q}(l)

Key Result

Theorem 1.1

Let $t, k, N \in \mathbb{Z}_{\geq 1}$ and $Q(x)$ be a polynomial of degree less than $\phi(N)k$ satisfying $Q(0) = 0$ and where $\phi(N)$ denotes Euler's totient function. Then, we have the following:

Theorems & Definitions (27)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 17 more