Power-Partible Reduction and Congruences for Apéry Numbers
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910525355458560 |
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| author | Wang, Rong-Hua Zhong, Michael X. X. |
| author_facet | Wang, Rong-Hua Zhong, Michael X. X. |
| contents | In this paper, we introduce the power-partible reduction for holonomic (or, P-recursive) sequences and apply it to obtain a series of congruences for Apéry numbers $A_k$. In particular, we prove that, for any $r\in\mathbb{N}$, there exists an integer $\tilde{c}_r$ such that \begin{equation*} \sum_{k=0}^{p-1}(2k+1)^{2r+1}A_k\equiv \tilde{c}_r p \pmod {p^3} \end{equation*} holds for any prime $p>3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01985 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Power-Partible Reduction and Congruences for Apéry Numbers Wang, Rong-Hua Zhong, Michael X. X. Combinatorics Number Theory 05A10, 11A07, 33F10 In this paper, we introduce the power-partible reduction for holonomic (or, P-recursive) sequences and apply it to obtain a series of congruences for Apéry numbers $A_k$. In particular, we prove that, for any $r\in\mathbb{N}$, there exists an integer $\tilde{c}_r$ such that \begin{equation*} \sum_{k=0}^{p-1}(2k+1)^{2r+1}A_k\equiv \tilde{c}_r p \pmod {p^3} \end{equation*} holds for any prime $p>3$. |
| title | Power-Partible Reduction and Congruences for Apéry Numbers |
| topic | Combinatorics Number Theory 05A10, 11A07, 33F10 |
| url | https://arxiv.org/abs/2301.01985 |