Some $q$-congruences with parameters

TL;DR

This paper uses a novel microscoping method to prove and generalize $q$-supercongruences involving cyclotomic polynomials, confirming conjectures and extending known results in the field.

Contribution

It applies a creative microscoping approach to verify conjectures on $q$-supercongruences modulo $\

Findings

Confirmed conjectures on $q$-supercongruences modulo $\

Provided parameter-generalizations of known $q$-supercongruences

Extended a $q$-analogue of a supercongruence of Rodriguez-Villegas

Abstract

Let $\Phi_n(q)$ be the $n$-th cyclotomic polynomial in $q$. Recently, the author and Zudilin provide a creative microscoping method to prove some $q$-supercongruences mainly modulo $\Phi_n(q)^3$ by introducing an additional parameter $a$. In this paper, we use this creative microscoping method to confirm some conjectures on $q$-supercongruences modulo $\Phi_n(q)^2$. We also give some parameter-generalizations of known $q$-supercongruences. For instance, we present further generalizations of a $q$-analogue of a famous supercongruence of Rodriguez-Villegas: $$ \sum_{k=0}^{p-1}\frac{{2k\choose k}^2}{16^k} \equiv (-1)^{(p-1)/2}\pmod{p^2}\quad\text{for any odd prime $p$.} $$