A Generalization of MacMahon Series via Cyclotomic Polynomials

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.