Some supercongruences on truncated hypergeometric series

Abstract

Abstract. In 2003, Rodriguez-Villegas conjectured four supercongruences on the truncated 3 F 2 hypergeometric series for certain modular K3 surfaces, which were gradually proved by several authors. Motivated by some supercongruences on combinatorial numbers such as Apéry numbers and Domb numbers, we establish some new supercongruences on the truncated 3 F 2 hypergeometric series, which extend the four Rodriguez-Villegas supercongruences.

Another Family of q-Congruences Modulo the Square of a Cyclotomic Polynomial

“…≡ 0 (mod p) for (p − 1)/2 < k p − 1. Nowadays all kinds of generalizations of (1.1) for p ≡ 3 (mod 4) are given by different authors [2,10,12,13,17,18,21,23,24]. For instance, Liu [17] proved that, for any prime p ≡ 3 (mod 4) and positive integer m,…”

Dwork-type supercongruences through a creative q-microscope

“…Liu [19] established the following generalization of (1.1): for any odd prime p and positive integer m,…”

A family of <inline-formula><tex-math id="M1">$ q $</tex-math></inline-formula>-congruences modulo the square of a cyclotomic polynomial

“…≡ 0 (mod p) for (p − 1)/2 < k ≤ p − 1. Nowadays various generalizations of (1.1) can be found in [8,10,11,12,13,14,16,17]. For example, Liu [12] proved that, for any prime p ≡ 3 (mod 4) and positive integer m,…”