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arXiv · 2412.14344

Euler-type recurrences for $t$-color and $t$-regular partition functions

Tapas Bhowmik·Wei-Lun Tsai·Dongxi Ye
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Abstract

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 $(q;q)_{\infty}^3.$

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Tapas Bhowmik, Wei-Lun Tsai, Dongxi Ye. 2024-12-18. Euler-type recurrences for $t$-color and $t$-regular partition functions. https://arxiv.org/abs/2412.14344

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