Some $q$-congruences with parameters
Volume 190 / 2019
Acta Arithmetica 190 (2019), 381-393
MSC: 33D15, 11A07, 11B65.
DOI: 10.4064/aa180624-16-12
Published online: 29 July 2019
Abstract
Let $\varPhi_n(q)$ be the $n$th cyclotomic polynomial in $q$. Recently, the author and Zudilin devised a method, called ‘creative microscoping’, to prove some $q$-supercongruences mainly modulo $\varPhi_n(q)^3$ by introducing an additional parameter $a$. In this paper, we use this method to confirm some conjectures on $q$-supercongruences modulo $\varPhi_n(q)^2$. 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:
