Supercongruences involving dual sequences

Abstract

In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\ldots$ define $$d_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k$$ and $$s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom nk(-1)^k\binom xk\binom{-1-x}k.$$ For any odd prime $p$ and $p$-adic integer $x$, we determine $\sum_{k=0}^{p-1}(\pm1)^kd_k(x)^2$ and $\sum_{k=0}^{p-1}(2k+1)d_k(x)^2$ modulo $p^2$; for example, we establish the new $p$-adic congruence $$\sum_{k=0}^{p-1}(-1)^kd_k(x)^2\equiv(-1)^{\langle x\rangle_p}\pmod{p^2},$$ where $\langle x\rangle_p$ denotes the least nonnegative integer $r$ with $x\equiv r\pmod p$. For any prime $p>3$ and $p$-adic integer $x$, we determine $\sum_{k=0}^{p-1}s_k(x)^2$ modulo $p^2$ (or $p^3$ if $x\in\{0,\ldots,p-1\}$), and show that $$\sum_{k=0}^{p-1}(2k+1)s_k(x)^2\equiv0\pmod{p^2}.$$ We also pose several related conjectures.