Congruences concerning Legendre polynomials

Abstract

Let $p$ be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for ${\sum }_{k=0}^{\frac{p-1}{2}}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2{m}^{-k}\mathrm{mod}{p}^2$. In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.

16 pages