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

TL;DR

This paper introduces a unified q-analogue of Van Hamme's supercongruences (C.2) and (G.2) using the q-Zeilberger algorithm, refining previous results and linking to Bernoulli number supercongruences.

Contribution

The authors develop a new q-analogue of key supercongruences, extending Van Hamme's results and providing a refined framework for related supercongruences.

Findings

Established a unified q-analogue of Van Hamme's supercongruences (C.2) and (G.2).

Derived a q-analogue of supercongruence involving Bernoulli numbers.

Provided a refinement of the (G.2) supercongruence.

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.