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

TL;DR

The paper resolves conjectural supercongruences for the Apéry-like sequence $t_n(x)$ by leveraging a $p$-adic expansion of binomial products, Beta-function representations, and Pfaff transformations. It derives explicit modulo $p^2$ formulas for $\sum_{n=0}^{p-1} t_n(x)^2$ and $\sum_{n=0}^{p-1}(n+1)t_n(x)^2$ in terms of $m=\langle x\rangle_p$ and Legendre symbols, and confirms Sun's conjecture in this setting. The results hinge on a careful partition of summation ranges, symmetry $t_n(x)=t_n(-1-x)$, and key summation identities that reduce complex double sums to closed forms. A corollary extends the findings to the case $x=-\tfrac12$, aligning with Sun's broader conjectures and strengthening the understanding of binomial-sum supercongruences for $p$-adic parameters.

Abstract

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.

Theorems & Definitions (21)

• Theorem 1.1
• Conjecture 1.1
• Theorem 1.2
• Corollary 1.1
• Lemma 2.1: Wang and Wang Wang2025
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4