Congruences concerning Legendre polynomials III

TL;DR

This paper explores congruences involving Legendre polynomials and binomial coefficients modulo primes, proving new identities and solving conjectures related to sums of binomial coefficients over finite fields.

Contribution

It establishes new congruences connecting Legendre polynomials with binomial sums and solves existing conjectures on these sums modulo prime squares.

Findings

Derived new congruences for Legendre polynomials modulo p

Connected binomial coefficient sums to Legendre polynomial evaluations

Solved conjectures of Sun and the author on binomial sums modulo p^2

Abstract

Let $p>3$ be a prime, and let $R_p$ be the set of rational numbers whose denominator is coprime to $p$. Let $\{P_n(x)\}$ be the Legendre polynomials. In this paper we mainly show that for $m,n,t\in R_p$ with $m\not\e 0\pmod p$, $$\align &P_{[\frac p6]}(t) \e -\Big(\frac 3p\Big)\sum_{x=0}^{p-1}\Big(\frac{x^3-3x+2t}p\Big)\pmod p, &\Big(\sum_{x=0}^{p-1}\Big(\frac{x^3+mx+n}p\Big)\Big)^2\equiv \Big(\frac{-3m}p\Big) \sum_{k=0}^{[p/6]}\binom{2k}k\binom{3k}k\binom{6k}{3k} \Big(\frac{4m^3+27n^2}{12^3\cdot 4m^3}\Big)^k\pmod p,$$ where $(\frac ap)$ is the Legendre symbol and $[x]$ is the greatest integer function. As an application we solve some conjectures of Z.W. Sun and the author concerning $\sum_{k=0}^{p-1}\binom{2k}k\binom{3k}k\binom{6k}{3k}/m^k\pmod {p^2}$, where $m$ is an integer not divisible by $p$.