Related papers: A Generalization of MacMahon Series via Cyclotomic…

In the present work, we extend current research in a nearly-forgotten but newly revived topic, initiated by P. A. MacMahon, on a generalized notion which relates the divisor sums to the theory of integer partitions and two infinite families…

We give explicit expressions for MacMahon's generalized sums-of-divisors $q$-series $A_r$ and $C_r$ by relating them to (odd) multiple Eisenstein series. Recently, these sums-of-divisors have been studied in the context of quasimodular…

Motivated by earlier work of P.~A.~MacMahon and recent contributions of T.~Amdeberhan, G.~E.~Andrews, K.~Ono, A.~Singh, and R.~Tauraso on higher-order partition enumerants, we study a class of $q$-series arising from nested divisor…

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}…

In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so we reveal a wealth of divisibility theorems and…

The quasisymmetic Macdonald polynomials $G_{\gamma}(X; q, t)$ were recently introduced by the first and second authors with Haglund, Mason, and Williams in [3] to refine the symmetric Macdonald polynomials $P_{\lambda}(X; q, t)$ with the…

In recent years, the generalized sum-of-divisor functions of MacMahon have been unified into the algebraic framework of $q$-multiple zeta values. In particular, these results link partition theory, quasimodular forms, $q$-multiple zeta…

We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters $(q,t)$ and polynomial in a further two parameters $(u,v)$. We evaluate these polynomials explicitly as a matrix…

Recently, MacMahon's generalized sum-of-divisor functions were shown to link partitions, quasimodular forms, and q-multiple zeta values. In this paper, we explore many further properties and extensions of these. Firstly, we address a…

Cyclotomy, the study of cyclotomic classes and cyclotomic numbers, is an area of number theory first studied by Gauss. It has natural applications in discrete mathematics and information theory. Despite this long history, there are…

Here we investigate the $q$-series \begin{align*} \mathcal{U}_a(q)&=\sum_{n=0}^{\infty} MO(a;n)q^n&:=\sum_{0< k_1<k_2<\cdots<k_a} \frac{q^{k_1+k_2+\cdots+k_a}}{(1-q^{k_1})^2(1-q^{k_2})^2\cdots(1-q^{k_a})^2},\\…

In [4] and [5], Folsom presents a family of modular units as higher-level analogues of the Rogers-Ramanujan $q$-continued fraction. These units are constructed from analytic solutions to the higher-order $q$-recurrence equations of Selberg.…

A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$. We prove that the degree of a $q$-holonomic sequence is eventually a quadratic…

The goal of this article is to study some basic algebraic and combinatorial properties of "generalized $n$-series" over a commutative ring $R$, which are functions $s: \mathbf{Z}_{\geq 0} \to R$ satisfying a mild condition. A special…

We prove that the ideal used in recent works to categorify the cyclotomic integers is generated by a cyclotomic polynomial. Moreover, we publish a proof by T. Ekedahl that the $q$-binomial relations used in the tensor product of…

In 1920, MacMahon introduced two families of $q$-series to study divisor sums. Recent work has shown that MacMahon's $q$-series are closely connected to overpartitions and $3$-colored partitions. Merca introduced truncated forms of…

Bachmann proves an identity expressing the generating series of MacMahon's generalized sum-of-divisors $q$-series in terms of Eisenstein series. MacMahon's $q$-series can be regarded as a $q$-analogue of the multiple zeta value $\zeta(2, 2,…

One variant of the $q$-Catalan polynomials is defined in terms of Gaussian polynomials by $\mathcal{C}_k(q)=\genfrac{[}{]}{0pt}{}{2k}{k}_q-q\genfrac{[}{]}{0pt}{}{2k}{k+1}_q$. Liu studied congruences of the form $\sum_{k=0}^{n-1}…

MacMahon's partition functions and their extensions provide equations that identify prime numbers as solutions. These results depend on the theory of (mixed weight) quasimodular forms on $SL_2(\mathbb{Z})$. Two of the authors, along with…

Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, we define a new polynomial related to the higher-order generalized…