Symbolic summation methods and hypergeometric supercongruences

Abstract

In this paper, via the WZ method and the summation package Sigma, we establish the following two supercongruences:∑k=0(p+1)/2(3k−1)(−12)k2(12)k4kk!3≡p−6p3(−1p)+2p3(−1p)Ep−3(modp4),∑k=0p−1(3k−1)(−12)k2(12)k4kk!3≡p−2p3(modp4), where p>3 is a prime, Ep−3 is the (p−3)-th Euler number and (−1p)=(−1)(p−1)/2 is the Legendre symbol. The first congruence modulo p3 confirms a recent conjecture of Guo and Schlosser.