A further $q$-generalization of the (C.2) and (G.2) supercongruences of Van Hamme

Abstract

Applying the $q$-Zeilberger algorithm, we establish a unified $q$-analogue of the (C.2) and (G.2) supercongruences of Van Hamme, which can be viewed as a refinement of several previously known results. As consequences, we obtain a $q$-analogue of supercongruence involving Bernoulli numbers, as well as a refinement of (G.2) supercongruence.