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

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.