Congruences concerning Legendre polynomials II

TL;DR

This paper proves conjectures related to sums involving binomial coefficients modulo prime squares and explores properties of Legendre polynomials at specific points, advancing understanding of these number theoretic functions.

Contribution

It solves several conjectures of Z.W. Sun concerning binomial sum congruences and provides new identities and evaluations for Legendre polynomials at specific arguments.

Findings

Proved binomial sum congruences modulo p^2 and p.

Established explicit formulas for Legendre polynomial evaluations modulo p.

Confirmed multiple conjectures of Z.W. Sun.

Abstract

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^3/m^k\pmod{p^2}$, $\sum_{k=0}^{p-1}\binom{2k}k\b{4k}{2k}/m^k\pmod p$ and $\sum_{k=0}^{p-1}\binom{2k}k^2\b{4k}{2k}/m^k\pmod {p^2}.$ In particular, we show that $\sum_{k=0}^{\frac{p-1}{2}}\binom{2k}k^3\equiv 0\pmod {p^2}$ for $p\equiv 3,5,6\pmod 7$. Let $P_n(x)$ be the Legendre polynomials. In the paper we also show that $ P_{[\frac {p}{4}]}(t)\equiv -\big(\frac{-6}{p}\big)\sum_{x=0}^{p-1} \big(\frac{x^3-3/2(3t+5)x-9t-7}{p}\big)\pmod p$ and determine $P_{\frac{p-1}{2}}(\sqrt 2), P_{\frac{p-1}{2}}(\frac{3\sqrt 2}{4}), P_{\frac{p-1}{2}}(\sqrt {-3}),P_{\frac{p-1}{2}}(\frac{\sqrt 3}{2}), P_{\frac{p-1}{2}}(\sqrt {-63}), P_{\frac{p-1}{2}}(\frac {3\sqrt 7}{8}) \pmod p$, where $t$ is a rational $p-$integer, $[x]$ is the greatest integer not exceeding $x$ and $(\frac {a}{p})$ is the Legendre symbol. As consequences we determine $P_{[\frac {p}{4}]}(t)\pmod p$ in the cases $t=-5/3,-7/9,-65/63$ and confirm many conjectures of Z.W. Sun.