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

TL;DR

This paper develops parametric generalizations of $q$-supercongruences using creative microscoping and the Chinese remainder theorem, leading to proofs of conjectured supercongruences and new related conjectures.

Contribution

It introduces a novel method combining creative microscoping with polynomial techniques to prove and generalize $q$-supercongruences modulo the fourth power of cyclotomic polynomials.

Findings

Proved a complete $q$-analogue of Van Hamme's (J.2) supercongruence.

Established a $q$-analogue of a Ramanujan-type supercongruence.

Proposed conjectures including a $q$-supercongruence modulo the fifth power of a cyclotomic polynomial.

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.