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13 July 2020
 
  » arxiv » 1101.5386

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Generalized Legendre polynomials and related congruences modulo $p^2$
Zhi-Hong Sun ;
Date 27 Jan 2011
AbstractFor any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial $P_n(a,x)=sum_{k=0}^ninom akinom{-1-a}k(frac{1-x}2)^k$. 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 $inom{(p-1)/2}{[p/8]}$ and $inom{(p-1)/2}{[p/(12)]}mod{p^2}$, where $[x]$ is the greatest integer function.
Source arXiv, 1101.5386
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