A $q$-Dwork-type generalization of Rodriguez-Villegas' supercongruences

TL;DR

This paper extends Rodriguez-Villegas' supercongruences using the creative microscoping method to establish a $q$-Dwork-type generalization, confirming recent conjectures and advancing the understanding of supercongruences.

Contribution

It introduces a new $q$-analogue of Rodriguez-Villegas' supercongruences using the creative microscoping technique, confirming a recent conjecture.

Findings

Established a $q$-Dwork-type generalization of supercongruences.

Validated the conjecture proposed by Guo and Zudilin.

Enhanced the analytical tools for supercongruence proofs.

Abstract

Guo and Zudilin [Adv. Math. 346 (2019), 329--358] developed an analytical method, called `creative microscoping', to prove many supercongruences by establishing their $q$-analogues. In this paper, we apply this method to give a $q$-Dwork-type generalization of Rodriguez-Villegas' supercongruences, which was recently conjectured by Guo and Zudilin.