$q$-Supercongruences with parameters

TL;DR

This paper introduces new $q$-supercongruences with parameters using the microscoping method, expanding the understanding of congruences in $q$-series and cyclotomic polynomials.

Contribution

It develops novel $q$-supercongruences with parameters by applying the microscoping method and the Chinese remainder theorem, extending previous results in the field.

Findings

Derived a $q$-supercongruence with four parameters modulo a cyclotomic polynomial product.

Established a $q$-supercongruence with two parameters modulo a power of the $q$-integer and cyclotomic polynomial.

Utilized the microscoping method and Chinese remainder theorem to obtain new supercongruences.

Abstract

In terms of the creative microscoping method recently introduced by Guo and Zudilin [Adv. Math. 346 (2019), 329--358], we find a $q$-supercongruence with four parameters modulo $\Phi_n(q)(1-aq^n)(a-q^n)$, where $\Phi_n(q)$ denotes the $n$-th cyclotomic polynomial in $q$. Then we empoly it and the Chinese remainder theorem for coprime polynomials to derive a $q$-supercongruence with two parameters modulo $[n]\Phi_n(q)^3$, where $[n]=(1-q^n)/(1-q)$ is the $q$-integer.