Arithmetic properties of MacMahon-type sums of divisors: the odd case

TL;DR

This paper explores the arithmetic properties of MacMahon-type sums of divisors, extending previous work to odd integer sums and establishing new Ramanujan-like congruences for their coefficients.

Contribution

It generalizes the second family of MacMahon sums over odd integers and proves new Ramanujan-like congruences for the resulting series coefficients.

Findings

Established new infinite families of Ramanujan-like congruences

Extended MacMahon sums to include odd integer restrictions

Connected divisor sums with partition theory

Abstract

A century ago, P. A. MacMahon introduced two families of generating functions, $$ \sum_{1\leq n_1<n_2<\cdots<n_t}\prod_{k=1}^t\frac{q^{n_k}}{(1-q^{n_k})^2} \quad\text{ and } \sum_{\substack{1\leq n_1<n_2<\cdots<n_t\\ \text{$n_1,n_2,\dots,n_t$ odd}}}\prod_{k=1}^t\frac{q^{n_k}}{(1-q^{n_k})^2}, $$ which connect sum-of-divisors functions and integer partitions. These have recently drawn renewed attention. In particular, Amdeberhan, Andrews, and Tauraso extended the first family above by defining $$ U_t(a,q):=\sum_{1\leq n_1<n_2<\cdots<n_t}\prod_{k=1}^t\frac{q^{n_k}}{1+aq^{n_k}+q^{2n_k}} $$ for $a=0, \pm1, \pm2$ and investigated various properties, including some congruences satisfied by the coefficients of the power series representations for $U_t(a,q)$. These arithmetic aspects were subsequently expanded upon by the authors of the present work. Our goal here is to generalize the second family of generating functions, where the sums run over odd integers, and then apply similar techniques to show new infinite families of Ramanujan--like congruences for the associated power series coefficients.