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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.04006 |
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| _version_ | 1866915917305217024 |
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| author | Qi, Wei-Wei |
| author_facet | Qi, Wei-Wei |
| contents | In $2012$, Guillera and Zudilin established the following two supercongruences involving truncated Ramanujan-type series: for any odd prime $p>2$, \begin{align*} \sum_{n=0}^{p-1}\frac{(\frac{1}{2})_n(\frac{1}{3})_n(\frac{1}{4})_n(\frac{3}{4})_n}{(1)_n^5}(-1)^n\left(172n^2+75n+9\right)\left(\frac{27}{16}\right)^n\equiv 9p^2 \pmod{p^5}, \end{align*} and \begin{align*} \sum_{n=0}^{p-1}\frac{(\frac{1}{2})_n(\frac{1}{3})_n(\frac{2}{3})_n}{(1)_n^3}\left(11n+3\right)\left(\frac{27}{16}\right)^n\equiv 3p \pmod{p^3}, \end{align*} where $(a)_n=\prod_{k=0}^{n-1}(a+k)$ denotes the Pochhammer symbol (rising factorial). In this paper, we mainly apply the Wilf-Zeilberger (WZ) method and symbolic summation techniques to prove these two supercongruences. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_04006 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Proof of Two Supercongruences of Guillera and Zudilin Qi, Wei-Wei Combinatorics In $2012$, Guillera and Zudilin established the following two supercongruences involving truncated Ramanujan-type series: for any odd prime $p>2$, \begin{align*} \sum_{n=0}^{p-1}\frac{(\frac{1}{2})_n(\frac{1}{3})_n(\frac{1}{4})_n(\frac{3}{4})_n}{(1)_n^5}(-1)^n\left(172n^2+75n+9\right)\left(\frac{27}{16}\right)^n\equiv 9p^2 \pmod{p^5}, \end{align*} and \begin{align*} \sum_{n=0}^{p-1}\frac{(\frac{1}{2})_n(\frac{1}{3})_n(\frac{2}{3})_n}{(1)_n^3}\left(11n+3\right)\left(\frac{27}{16}\right)^n\equiv 3p \pmod{p^3}, \end{align*} where $(a)_n=\prod_{k=0}^{n-1}(a+k)$ denotes the Pochhammer symbol (rising factorial). In this paper, we mainly apply the Wilf-Zeilberger (WZ) method and symbolic summation techniques to prove these two supercongruences. |
| title | Proof of Two Supercongruences of Guillera and Zudilin |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.04006 |