On Glaisher's Partition Theorem
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917400823201792 |
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| author | Andrews, George E. Dhar, Aritram |
| author_facet | Andrews, George E. Dhar, Aritram |
| contents | Glaisher's theorem states that the number of partitions of $n$ into parts which repeat at most $m-1$ times is equal to the number of partitions of $n$ into parts which are not divisible by $m$. The $m=2$ case is Euler's famous partition theorem. Recently, Andrews, Kumar, and Yee gave two new partition functions $C(n)$ and $D(n)$ related to Euler's theorem. Lin and Zang extended their result to Glaisher's theorem by generalizing $C(n)$. We generalize $D(n)$ and prove an analogous partition identity for the $m=3$ case. We also provide a new series equal to Glaisher's product both in the finite and infinite cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Glaisher's Partition Theorem Andrews, George E. Dhar, Aritram Combinatorics Number Theory 05A17, 05A19, 11P81, 11P84 Glaisher's theorem states that the number of partitions of $n$ into parts which repeat at most $m-1$ times is equal to the number of partitions of $n$ into parts which are not divisible by $m$. The $m=2$ case is Euler's famous partition theorem. Recently, Andrews, Kumar, and Yee gave two new partition functions $C(n)$ and $D(n)$ related to Euler's theorem. Lin and Zang extended their result to Glaisher's theorem by generalizing $C(n)$. We generalize $D(n)$ and prove an analogous partition identity for the $m=3$ case. We also provide a new series equal to Glaisher's product both in the finite and infinite cases. |
| title | On Glaisher's Partition Theorem |
| topic | Combinatorics Number Theory 05A17, 05A19, 11P81, 11P84 |
| url | https://arxiv.org/abs/2512.12346 |