A further $q$-analogue of Van Hamme's (H.2) supercongruence for $p\equiv1\pmod{4}$

TL;DR

This paper develops a new $q$-analogue of a supercongruence related to Van Hamme's (H.2) for primes congruent to 1 modulo 4, extending previous work on supercongruences and their $q$-analogues.

Contribution

It introduces a novel $q$-analogue of the Long--Ramakrishna formula specifically for primes $p \equiv 1 \pmod{4}$ using $q$-Whipple formulas and the Chinese remainder theorem.

Findings

Derived a $q$-analogue for $p \equiv 1 \pmod{4}$

Extended the supercongruence to the $q$-setting

Connected $q$-series identities with number theory

Abstract

Several years ago, Long and Ramakrishna [Adv. Math. 290 (2016), 773--808] extended Van Hamme's (H.2) supercongruence to the modulus $p^3$ case. Recently, Guo [Int. J. Number Theory, to appear] found a $q$-analogue of the Long--Ramakrishna formula for $p\equiv 3\pmod 4$. In this note, a $q$-analogue of the Long--Ramakrishna formula for $p\equiv 1\pmod 4$ is derived through the $q$-Whipple formulas and the Chinese remainder theorem for coprime polynomials.