$q$-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping

Abstract

By applying Chinese remainder theorem for coprime polynomials and the "creative microscoping" method recently introduced by the author and Zudilin, we establish parametric generalizations of three $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. The original $q$-supercongruences then follow from these parametric generalizations by taking the limits as the parameter tends to $1$ (l'H\^opital's rule is utilized here). In particular, we prove a complete $q$-analogue of the (J.2) supercongruence of Van Hamme and a complete $q$-analogue of a "divergent" Ramanujan-type supercongruence, thus confirming two recent conjectures of the author. We also put forward some related conjectures, including a $q$-supercongruence modulo the fifth power of a cyclotomic polynomial.