On some conjectural supercongruences involving the sequence $t_n(x)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908589406289920 |
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| author | Han, Hui-Li Wang, Chen |
| author_facet | Han, Hui-Li Wang, Chen |
| contents | In this paper, we study some supercongruences involving the sequence $$ t_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}2^k $$ and solve some open problems. For any odd prime $p$ and $p$-adic integer $x$, we determine $\sum_{n=0}^{p-1}t_n(x)^2$ and $\sum_{n=0}^{p-1}(n+1)t_n(x)^2$ modulo $p^2$; for example, we establish that
\begin{align*}
\sum_{n=0}^{p-1}t_n(x)^2\equiv\begin{cases}
\left(\dfrac{-1}{p}\right)\pmod{p^2},&\text{if }2x\equiv-1\pmod{p},\\[8pt]
(-1)^{\langle x\rangle_p}\dfrac{p+2(x-\langle x\rangle_p)}{2x+1}\pmod{p^2},&\text{otherwise,}
\end{cases}
\end{align*} where $\langle x\rangle_p$ denotes the least nonnegative residue of $x$ modulo $p$. This confirms a conjecture of Z.-W. Sun. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11338 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On some conjectural supercongruences involving the sequence $t_n(x)$ Han, Hui-Li Wang, Chen Number Theory Combinatorics In this paper, we study some supercongruences involving the sequence $$ t_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}2^k $$ and solve some open problems. For any odd prime $p$ and $p$-adic integer $x$, we determine $\sum_{n=0}^{p-1}t_n(x)^2$ and $\sum_{n=0}^{p-1}(n+1)t_n(x)^2$ modulo $p^2$; for example, we establish that \begin{align*} \sum_{n=0}^{p-1}t_n(x)^2\equiv\begin{cases} \left(\dfrac{-1}{p}\right)\pmod{p^2},&\text{if }2x\equiv-1\pmod{p},\\[8pt] (-1)^{\langle x\rangle_p}\dfrac{p+2(x-\langle x\rangle_p)}{2x+1}\pmod{p^2},&\text{otherwise,} \end{cases} \end{align*} where $\langle x\rangle_p$ denotes the least nonnegative residue of $x$ modulo $p$. This confirms a conjecture of Z.-W. Sun. |
| title | On some conjectural supercongruences involving the sequence $t_n(x)$ |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2510.11338 |