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.
Categories: math.NT math.CO