Generalized Legendre polynomials and related supercongruences

For any positive integer n and variables a and x   we define the generalized Legendre polynomial Pn(a,x)Pn(a,x) by Pn(a,x)=∑k=0n(ak)(−1−ak)(1−x2)k. Let p   be an odd prime. In this paper we prove many congruences modulo p2p2 related to Pp−1(a,x)Pp−1(a,x). For example, we show that Pp−1(a,x)≡(−1)〈a〉pPp−1(a,−x)(modp2), where a is a rational p  -adic integer and 〈a〉p〈a〉p is the least nonnegative residue of a modulo p  . We also generalize some congruences of Zhi-Wei Sun, and establish congruences for ∑k=0p−1(2kk)(3kk)/54k and ∑k=0p−1(ak)(b−ak)(modp2).