Congruences concerning Legendre polynomials

TL;DR

This paper explores congruences involving Legendre polynomials and binomial coefficients modulo prime squares, confirming some conjectures and proposing new ones related to supercongruences.

Contribution

It proves new congruences for sums involving Legendre polynomials and binomial coefficients modulo p^2, confirming several of Z.W. Sun's conjectures and proposing 13 new conjectures.

Findings

Confirmed several conjectures of Z.W. Sun

Derived new congruences for binomial sums involving Legendre polynomials

Proposed 13 new conjectures on supercongruences

Abstract

Let $p$ be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for $\sum_{k=0}^{\frac{p-1}2}\binom{2k}k^2m^{-k}\mod {p^2}$. In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.