Euler-type recurrences for $t$-color and $t$-regular partition functions
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| Format: | Preprint |
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2024
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| _version_ | 1866917875945570304 |
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| author | Bhowmik, Tapas Tsai, Wei-Lun Ye, Dongxi |
| author_facet | Bhowmik, Tapas Tsai, Wei-Lun Ye, Dongxi |
| contents | We give Euler-like recursive formulas for the $t$-colored partition function when $t=2$ or $t=3,$ as well as for all $t$-regular partition functions. In particular, we derive an infinite family of ``triangular number" recurrences for the $3$-colored partition function. Our proofs are inspired by the recent work of Gomez, Ono, Saad, and Singh on the ordinary partition function and make extensive use of $q$-series identities for $(q;q)_{\infty}$ and $(q;q)_{\infty}^3.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_14344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Euler-type recurrences for $t$-color and $t$-regular partition functions Bhowmik, Tapas Tsai, Wei-Lun Ye, Dongxi Number Theory Combinatorics 05A17, 11F11, 11F25, 11P81 We give Euler-like recursive formulas for the $t$-colored partition function when $t=2$ or $t=3,$ as well as for all $t$-regular partition functions. In particular, we derive an infinite family of ``triangular number" recurrences for the $3$-colored partition function. Our proofs are inspired by the recent work of Gomez, Ono, Saad, and Singh on the ordinary partition function and make extensive use of $q$-series identities for $(q;q)_{\infty}$ and $(q;q)_{\infty}^3.$ |
| title | Euler-type recurrences for $t$-color and $t$-regular partition functions |
| topic | Number Theory Combinatorics 05A17, 11F11, 11F25, 11P81 |
| url | https://arxiv.org/abs/2412.14344 |