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

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.