Proof of two supercongruences of truncated hypergeometric series ${}_4F_3$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866910705668587520 |
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| author | Mao, Guo-Shuai |
| author_facet | Mao, Guo-Shuai |
| contents | In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{(p-1)/2}\frac{6n+1}{(-512)^n}\binom{2n}n^3&\equiv p\left(\frac{-2}p\right)+\frac{p^3}4\left(\frac2p\right)E_{p-3}\pmod{p^4}, \end{align*} where $\left(\frac{\cdot}p\right)$ stands for the Legendre symbol, and $E_{n}$ is the $n$-th Euler number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_09983 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Proof of two supercongruences of truncated hypergeometric series ${}_4F_3$ Mao, Guo-Shuai Number Theory Combinatorics In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{(p-1)/2}\frac{6n+1}{(-512)^n}\binom{2n}n^3&\equiv p\left(\frac{-2}p\right)+\frac{p^3}4\left(\frac2p\right)E_{p-3}\pmod{p^4}, \end{align*} where $\left(\frac{\cdot}p\right)$ stands for the Legendre symbol, and $E_{n}$ is the $n$-th Euler number. |
| title | Proof of two supercongruences of truncated hypergeometric series ${}_4F_3$ |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/1910.09983 |