Legendre polynomials and supercongruences
Volume 159 / 2013
Abstract
Let $p>3$ be a prime, and let $R_p$ be the set of rational numbers whose denominator is not divisible by $p$. Let $\{P_n(x)\}$ be the Legendre polynomials. In this paper we mainly show that for $m,n,t\in R_p$ with $m\not\equiv 0\pmod p$, $$ P_{[\frac p6]}(t) \equiv -\biggl(\frac 3p\biggr)\sum_{x=0}^{p-1}\biggl(\frac{x^3-3x+2t}p\biggr)\pmod p $$ and $$ \biggl(\sum_{x=0}^{p-1}\biggl(\frac{x^3+mx+n}p\biggr)\biggr)^2 \equiv \biggl(\frac{-3m}p\biggr) \sum_{k=0}^{[p/6]}\binom{2k}k\binom{3k}k\binom{6k}{3k} \biggl(\frac{4m^3+27n^2}{12^3\cdot 4m^3}\biggr)^k\pmod p, $$ 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 $p$.
