A $q$-supercongruence modulo the fourth power of a cyclotomic polynomial

TL;DR

This paper establishes a new $q$-supercongruence involving two parameters modulo the fourth power of a cyclotomic polynomial, utilizing advanced hypergeometric transformations and the Chinese remainder theorem.

Contribution

It introduces a novel $q$-supercongruence with parameters, expanding the understanding of supercongruences in the context of cyclotomic polynomials.

Findings

Derived $q$-supercongruence with two free parameters

Established congruences involving Bernoulli numbers

Applied Watson's $_8 ext{phi}_7$ transformation and microscoping method

Abstract

In this paper, a new $q$-supercongruence with two free parameters modulo the fourth power of a cyclotomic polynomial is obtained. Our main auxiliary tools are Watson's $_8\phi_7$ transformation formula for basic hypergeometric series, the `creative microscoping' method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials. By taking suitable parameter substitutions in the established $q$-supercongruence, some nice congruences involving the Bernoulli numbers are derived.