On a conjectural supercongruence involving the dual sequence $s_n(x)$

TL;DR

This paper proves a conjecture by Z.-W. Sun regarding a supercongruence involving the sequence of polynomials s_n(x), extending understanding of their congruence properties modulo prime powers.

Contribution

The paper confirms Sun's conjecture on a supercongruence for the sequence s_n(x), providing a significant advancement in the study of polynomial congruences and supercongruences.

Findings

Confirmed Sun's supercongruence conjecture for all primes p>3.

Established explicit congruence relations for the sequence s_n(x).

Extended the understanding of polynomial supercongruences in number theory.

Abstract

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.