Some supercongruences on truncated ${}_3F_2$ hypergeometric series

Abstract

In 2003, Rodriguez-Villegas conjectured four supercongruences on the truncated ${}_3F_2$ hypergeometric series for certain modular K3 surfaces, which were gradually proved by several authors. Motivated by some supercongruences on combinatorial numbers such as Ap\'ery numbers and Domb numbers, we establish some new supercongruences on the truncated ${}_3F_2$ hypergeometric series, which extend the four Rodriguez-Villegas supercongruences.