The Legendre polynomials \(P_n(x)\) are defined by

$$\begin{aligned} P_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n+k\\ k\end{array}}\right) \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( \frac{x-1}{2}\right) ^k\quad (n=0,1,2,\ldots ). \end{aligned}$$

In this paper, we prove two congruences concerning Legendre polynomials. For any prime \(p>3\), by using the symbolic summation package Sigma, we show that

$$\begin{aligned} \sum _{k=0}^{p-1}(2k+1)P_k(-5)^3\equiv p-\frac{10}{3}p^2q_p(2)\pmod {p^3}, \end{aligned}$$

where \(q_p(2)=(2^{p-1}-1)/p\) is the Fermat quotient. This confirms a conjecture of Z.-W. Sun. Furthermore, we prove the following congruence which was conjectured by V.J.W. Guo

$$\begin{aligned}&\sum _{k=0}^{p-1}(-1)^k(2k+1)P_k(2x+1)^4\\ \equiv&p\sum _{k=0}^{(p-1)/2}(-1)^k\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2(x^2+x)^k(2x+1)^{2k}\pmod {p^3},\end{aligned}$$

where p is an odd prime and x is an integer.

中文翻译：

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有关勒让德多项式的两个同余证明

勒让德多项式\（P_n（x）\）由下式定义

$$ \ begin {aligned} P_n（x）= \ sum _ {k = 0} ^ n \ left（{\ begin {array} {c} n + k \\ k \ end {array}} \ right）\左（{\ begin {array} {c} n \\ k \ end {array}} \ right）\ left（\ frac {x-1} {2} \ right）^ k \ quad（n = 0,1 ，2，\ ldots）。\ end {aligned} $$

在本文中，我们证明了勒让德多项式的两个等价性。对于任何素数\（p> 3 \），通过使用符号求和包Sigma，我们得出

$$ \ begin {aligned} \ sum _ {k = 0} ^ {p-1}（2k + 1）P_k（-5）^ 3 \ equiv p- \ frac {10} {3} p ^ 2q_p（2 ）\ pmod {p ^ 3}，\ end {aligned} $$

其中\（q_p（2）=（2 ^ {p-1} -1）/ p \）是Fermat商。这证实了Z.-W.的猜想。太阳。此外，我们证明了以下由VJW Guo猜想的全等

$$ \ begin {aligned}＆\ sum _ {k = 0} ^ {p-1}（-1）^ k（2k + 1）P_k（2x + 1）^ 4 \\ \ equiv＆p \ sum _ {k = 0} ^ {（p-1）/ 2}（-1）^ k \ left（{\ begin {array} {c} 2k \\ k \ end {array}} \ right）^ 2（x ^ 2 + x）^ k（2x + 1）^ {2k} \ pmod {p ^ 3}，\ end {aligned} $$

其中p是奇数质数，x是整数。