Proof of Two Congruences Concerning Legendre Polynomials

The Legendre polynomials P-n(x) are defined by P-n(x) = Sigma(n)(k=0) ((n + k)(k)) ((n)(k)) (x - 1/2)(k) (n = 0, 1, 2, ... ). 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 Sigma(p-1)(k=0)(2k+1)P-k(-5)(3) equivalent to p - 10/3p(2)qp(2)(mod p(3)), where qp(()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 Sigma(p-1)(k=0)(-1)(k)(2k+1)P-k(2x+1)(4) equivalent to p Sigma((p-1)/2)(k=0)(-1)(k)((2k)(k))(2) (x(2) + x)(k)(2x + 1)(2k) (mod p(3)), where p is an odd prime and x is an integer.