On some conjectural supercongruences involving the sequence $t_n(x)$

Abstract

In this paper, we study some supercongruences involving the sequence $$t_n(x)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{x}{k}\right)\left(\genfrac{}{}{0ex}{}{x+k}{k}\right)2^k$$ 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 $⟨x{⟩}_p$ denotes the least nonnegative residue of $x$ modulo $p$. This confirms a conjecture of Z.-W. Sun.

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