Congruences for the Apéry numbers modulo $p^3$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913546686693376 |
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| author | Sun, Zhi-Hong |
| author_facet | Sun, Zhi-Hong |
| contents | Let $\{A'_n\}$ be the Apéry numbers given by $A'_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k.$ For any prime $p\equiv 3\pmod 4$ we show that $A'_{\frac{p-1}2}\equiv \frac{p^2}3\binom{\frac{p-3}2}{\frac{p-3}4}^{-2}\pmod {p^3}$. Let $\{t_n\}$ be given by $$t_0=1,\ t_1=5\quad\hbox{and}\quad t_{n+1}=(8n^2+12n+5)t_n-4n^2(2n+1)^2t_{n-1}\ (n\ge 1).$$ We also obtain the congruences for $t_p\pmod {p^3},\ t_{p-1}\pmod {p^2}$ and $t_{\frac{p-1}2}\pmod {p^2}$, where $p$ is an odd prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06544 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Congruences for the Apéry numbers modulo $p^3$ Sun, Zhi-Hong Number Theory Combinatorics Primary 11A07, Secondary 05A10, 05A19, 11B68, 11E25 Let $\{A'_n\}$ be the Apéry numbers given by $A'_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k.$ For any prime $p\equiv 3\pmod 4$ we show that $A'_{\frac{p-1}2}\equiv \frac{p^2}3\binom{\frac{p-3}2}{\frac{p-3}4}^{-2}\pmod {p^3}$. Let $\{t_n\}$ be given by $$t_0=1,\ t_1=5\quad\hbox{and}\quad t_{n+1}=(8n^2+12n+5)t_n-4n^2(2n+1)^2t_{n-1}\ (n\ge 1).$$ We also obtain the congruences for $t_p\pmod {p^3},\ t_{p-1}\pmod {p^2}$ and $t_{\frac{p-1}2}\pmod {p^2}$, where $p$ is an odd prime. |
| title | Congruences for the Apéry numbers modulo $p^3$ |
| topic | Number Theory Combinatorics Primary 11A07, Secondary 05A10, 05A19, 11B68, 11E25 |
| url | https://arxiv.org/abs/2409.06544 |