Some $q$-supercongruences modulo the fourth power of a cyclotomic polynomial

TL;DR

This paper proves a new $q$-supercongruence involving cyclotomic polynomials using the creative microscoping method, confirming a recent conjecture and generalizing previous supercongruences in the field.

Contribution

It introduces a novel $q$-supercongruence modulo $[n]\

Findings

Established a $q$-supercongruence modulo $[n]\

Confirmed a recent conjecture of Guo,

Generalized previous $q$-supercongruences by Guo and Schlosser.

Abstract

In terms of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a $q$-supercongruence with two parameters modulo $[n]\Phi_n(q)^3$. Here $[n]=(1-q^n)/(1-q)$ and $\Phi_n(q)$ is the $n$-th cyclotomic polynomial in $q$. In particular, we confirm a recent conjecture of Guo and give a complete $q$-analogue of Long's supercongruence. The latter is also a generalization of a recent $q$-supercongruence obtained by Guo and Schlosser.