Extensions of MacMahon's sums of divisors

TL;DR

This paper extends MacMahon's divisor sums using rational function approximations, revealing new divisibility theorems and combinatorial identities related to q-harmonic sums and partition theory.

Contribution

It introduces a novel approach to generalize MacMahon's sums through rational function approximation, uncovering new divisibility properties and identities in partition and q-series theory.

Findings

Revealed new divisibility theorems for divisor sums.

Discovered unexpected combinatorial identities involving q-harmonic sums.

Extended MacMahon's sums with a different analytical approach.

Abstract

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 unexpected combinatorial identities. Our initial approach is quite different from MacMahon and involves rational function approximation to MacMahon-type generating functions. One such example involves multiple $q$-harmonic sums $$\sum_{k=1}^n\frac{(-1)^{k-1}\genfrac{[}{]}{0pt}{}{n}{k}_{q}(1+q^k)q^{\binom{k}{2}+tk}}{[k]_q^{2t} \genfrac{[}{]}{0pt}{}{n+k}{k}_{q}}=\sum_{1\leq k_1\leq\cdots\leq k_{2t}\leq n}\frac{q^{n+k_1+k_3\cdots+k_{2t-1}}+q^{k_2+k_4+\cdots+k_{2t}}}{[n+k_1]_q[k_2]_q\cdots[k_{2t}]_q}.$$