Some supercongruences on truncated ${}_3F_2$ hypergeometric series

TL;DR

This paper proves new supercongruences for truncated ${}_3F_2$ hypergeometric series, extending earlier conjectures related to modular K3 surfaces, inspired by supercongruences in combinatorial numbers.

Contribution

It establishes novel supercongruences on truncated ${}_3F_2$ hypergeometric series, expanding upon Rodriguez-Villegas's conjectures and connecting to combinatorial supercongruences.

Findings

New supercongruences on truncated ${}_3F_2$ series are proved.

Extensions of Rodriguez-Villegas's four supercongruences are demonstrated.

Connections to supercongruences in combinatorial numbers are explored.

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.