A refined $q$-analogue of some congruences of Van Hamme

TL;DR

This paper develops a unified q-analogue of certain supercongruences originally proposed by Van Hamme, refining previous results and establishing new parametric supercongruences linked to Euler polynomials.

Contribution

It introduces a novel unified q-analogue for specific Van Hamme supercongruences, extending and refining earlier findings in the field.

Findings

Established a unified q-analogue for Van Hamme's supercongruences (B.2), (E.2), (F.2).

Refined known supercongruences with a new parametric supercongruence.

Connected supercongruences to Euler polynomials.

Abstract

In 1997, Van Hamme proposed 13 supercongruences corresponding to $1/\pi$ series of the Ramanujan-type. Inspired by the recent work of V.J.W. Guo, we establish a unified $q$-analogue of Van Hamme's (B.2), (E.2) and (F.2) supercongruences, which is also a refinement of some known results obtained by different authors. As a consequence, we give a parametric supercongruence related to the Euler polynomials.