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arXiv · 2409.20400

Further study on MacMahon-type sums of divisors

Tewodros Amdeberhan·George E. Andrews·Roberto Tauraso
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Abstract

This paper is devoted to the study of $$ U_t(a,q):=\sum_{1\leq n_1<n_2<\cdots<n_t}\frac{q^{n_1+n_2+\cdots+n_t}}{(1+aq^{n_1}+q^{2n_1})(1+aq^{n_2}+q^{2n_2})\cdots(1+aq^{n_t}+q^{2n_t})} $$ when $a$ is one of $0, \pm 1, \pm2$. The idea builds on our previous treatment of the case $a=-2$. It is shown that all these functions lie in the ring of quasi-modular forms. Among the more surprising findings is $$U_2(1,q)=\sum_{n\geq1} \frac{q^{3n}}{(1-q^{3n})^2}.$$

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Tewodros Amdeberhan, George E. Andrews, Roberto Tauraso. 2024-09-30. Further study on MacMahon-type sums of divisors. https://arxiv.org/abs/2409.20400

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