Generalized Legendre polynomials and related congruences modulo $p^2$

Abstract

For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial $P_n(a,x)=\sum_{k=0}^n\b ak\b{-1-a}k(\frac{1-x}2)^k$. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related to $P_{p-1}(a,x)$. For example, we show that $P_{p-1}(a,x)\e (-1)^{<a>_p}P_{p-1}(a,-x)\mod {p^2}$, where $<a>_p$ is the least nonnegative residue of $a$ modulo $p$. We also generalize some congruences of Zhi-Wei Sun, and determine $\sum_{k=0}^{p-1}\binom{2k}k\binom{3k}k{54^{-k}}$ and $\sum_{k=0}^{p-1}\binom ak\binom{b-a}k\mod {p^2}$, where $[x]$ is the greatest integer function. Finally we pose some supercongruences modulo $p^2$ concerning binary quadratic forms.