Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

q-analogues of the G.2 Supercongruence of Van Hamme

Yudong Liu, Xiaoxia Wang

Abstract

Motivated by the recent research of congruences and $q$-congruences, we provide two different $q$-analogues of the (G.2) supercongruence of Van Hamme through the `creative microscoping' method, which was devised by Guo and Zudilin. It is a remarkable fact that this is the first time to give direct $q$-analogues of (G.2). In addition, we propose a conjecture related to Swisher's Dwork-type supercongruence (G.3).

q-analogues of the G.2 Supercongruence of Van Hamme

Abstract

Motivated by the recent research of congruences and -congruences, we provide two different -analogues of the (G.2) supercongruence of Van Hamme through the `creative microscoping' method, which was devised by Guo and Zudilin. It is a remarkable fact that this is the first time to give direct -analogues of (G.2). In addition, we propose a conjecture related to Swisher's Dwork-type supercongruence (G.3).

Paper Structure

This paper contains 5 sections, 9 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. Proof of Theorem 2
  4. Generalizations of Theorems \ref{['thm:1']} and \ref{['thm:2']}
  5. A Conjecture about Swisher's (G.3)

Key Result

Theorem 1

Let $n\equiv 1\pmod {4}$ be a positive integer. Then

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • proof : Proof of Theorem 1.
  • Lemma 3
  • ...and 9 more