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 ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\left(\genfrac{}{}{0ex}{}{3k}{k}\right)54^{-k}$ and ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{a}{k}\right)\left(\genfrac{}{}{0ex}{}{b-a}{k}\right)\mathrm{mod}{p}^2$, where $[x]$ is the greatest integer function. Finally we pose some supercongruences modulo $p^2$ concerning binary quadratic forms.

37 pages

References in corpus (6)

• Open Conjectures on Congruences
• New congruences involving products of two binomial coefficients
• On sums involving products of three binomial coefficients
• On sums of Apéry polynomials and related congruences
• Supercongruences for a truncated hypergeometric series
• Some supercongruences modulo $p^2$

Cited by in corpus (2)

• Some q-analogues of supercongruences of Rodriguez-Villegas
• On the divisibility of the truncated hypergeometric function 3F2