$p$-adic analogues of hypergeometric identities and their applications

TL;DR

This paper proves several conjectures related to $p$-adic hypergeometric identities, specifically confirming congruences involving Apéry numbers and primes, with applications to number theory and combinatorics.

Contribution

The paper establishes new $p$-adic hypergeometric identities and confirms conjectures posed by Sun, advancing understanding of Apéry numbers modulo prime squares.

Findings

Confirmed Sun's conjectures on Apéry numbers modulo $p^2$

Established $p$-adic hypergeometric identities

Derived congruences involving primes of specific forms

Abstract

In this paper, we confirm several conjectures posed by Sun recently; for example, we prove that for any odd prime $p$ we have $$ \sum_{k=0}^{p-1}A_k\equiv\begin{cases}4x^2-2p\pmod{p^2}\quad&\text{if $p=x^2+2y^2\ (x,y\in\mathbb{Z})$},\\ 0\pmod{p^2}\quad&\text{if $p\equiv5,7\pmod{8}$},\end{cases} $$ where $A_n:=\sum_{k=0}^n\binom{n+k}{k}^2\binom{n}{k}^2$ are the Ap\'{e}ry numbers.