On a conjectural supercongruence involving the dual sequence $s_n(x)$
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| Format: | Preprint |
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2025
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| _version_ | 1866912444584034304 |
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| author | Wang, Chen Wang, Sheng-Jie |
| author_facet | Wang, Chen Wang, Sheng-Jie |
| contents | In 2017, motivated by a supercongruence conjectured by Kimoto and Wakayama and confirmed by Long, Osburn and Swisher, Z.-W. Sun introduced the sequence of polynomials: $$ s_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}=\sum_{k=0}^n\binom{n}{k}(-1)^k\binom{x}{k}\binom{-1-x}{k} $$ and investigated its congruence properties. In particular, Z.-W. Sun conjectured that for any prime $p>3$ and $p$-adic integer $x\neq-1/2$ one has \begin{equation*} \sum_{n=0}^{p-1}s_n(x)^2\equiv (-1)^{\langle x\rangle_p}\frac{p+2(x-\langle x\rangle_p)}{2x+1}\pmod{p^3}, \end{equation*} where $\langle x\rangle_p$ denotes the least nonnegative residue of $x$ modulo $p$. In this paper, we confirm this conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_18287 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a conjectural supercongruence involving the dual sequence $s_n(x)$ Wang, Chen Wang, Sheng-Jie Number Theory Combinatorics In 2017, motivated by a supercongruence conjectured by Kimoto and Wakayama and confirmed by Long, Osburn and Swisher, Z.-W. Sun introduced the sequence of polynomials: $$ s_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}=\sum_{k=0}^n\binom{n}{k}(-1)^k\binom{x}{k}\binom{-1-x}{k} $$ and investigated its congruence properties. In particular, Z.-W. Sun conjectured that for any prime $p>3$ and $p$-adic integer $x\neq-1/2$ one has \begin{equation*} \sum_{n=0}^{p-1}s_n(x)^2\equiv (-1)^{\langle x\rangle_p}\frac{p+2(x-\langle x\rangle_p)}{2x+1}\pmod{p^3}, \end{equation*} where $\langle x\rangle_p$ denotes the least nonnegative residue of $x$ modulo $p$. In this paper, we confirm this conjecture. |
| title | On a conjectural supercongruence involving the dual sequence $s_n(x)$ |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2506.18287 |