New $q$-supercongruences arising from a summation of basic hypergeometric series

TL;DR

This paper introduces new $q$-supercongruences derived from basic hypergeometric series summations, employing the microscoping method and Chinese remainder theorem, including a $q$-analogue of Liu's formula.

Contribution

It presents novel $q$-supercongruences using advanced summation techniques and extends Liu's formula into the $q$-analogue domain.

Findings

Derived new $q$-supercongruences from hypergeometric series.

Established a $q$-analogue of Liu's formula.

Applied the microscoping method and Chinese remainder theorem.

Abstract

With the help of a summation of basic hypergeometric series, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we find some new $q$-supercongruences. Especially, we give a $q$-analogue of a formula due to Liu [J. Math. Anal. Appl. 497 (2021), Art.~124915].