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

TL;DR

This paper introduces the first direct $q$-analogues of Van Hamme's (G.2) supercongruence using the creative microscoping method, and proposes a related conjecture on Swisher's supercongruence.

Contribution

It provides the first direct $q$-analogues of the (G.2) supercongruence and introduces a new conjecture related to Swisher's Dwork-type supercongruence.

Findings

Two different $q$-analogues of (G.2) supercongruence are constructed.

The creative microscoping method is effectively applied to derive these $q$-analogues.

A new conjecture related to Swisher's supercongruence is proposed.

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).