Proof of Two Congruences Concerning Legendre Polynomials

Abstract

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.