Further study on MacMahon-type sums of divisors
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912052094697472 |
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| author | Amdeberhan, Tewodros Andrews, George E. Tauraso, Roberto |
| author_facet | Amdeberhan, Tewodros Andrews, George E. Tauraso, Roberto |
| contents | 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 $$U_2(1,q)=\sum_{n\geq1} \frac{q^{3n}}{(1-q^{3n})^2}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20400 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Further study on MacMahon-type sums of divisors Amdeberhan, Tewodros Andrews, George E. Tauraso, Roberto Number Theory Combinatorics 11M32, 11P83, 11F37 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 $$U_2(1,q)=\sum_{n\geq1} \frac{q^{3n}}{(1-q^{3n})^2}.$$ |
| title | Further study on MacMahon-type sums of divisors |
| topic | Number Theory Combinatorics 11M32, 11P83, 11F37 |
| url | https://arxiv.org/abs/2409.20400 |