$q$-Supercongruences from the $q$-Saalsch\"{u}tz identity

TL;DR

This paper develops new $q$-supercongruences using the $q$-Saalsch"{u}tz identity and Chinese remainder theorem, providing a $q$-analogue of a known formula and extending the theory of supercongruences.

Contribution

It introduces novel $q$-supercongruences modulo the third power of cyclotomic polynomials, expanding the understanding of $q$-series and supercongruences.

Findings

Established $q$-supercongruences modulo third power of cyclotomic polynomials.

Provided a $q$-analogue of a formula by Long and Ramakrishna.

Utilized $q$-Saalsch"{u}tz identity and Chinese remainder theorem in proofs.

Abstract

In terms of the $q$-Saalsch\"{u}tz identity and the Chinese remainder theorem for coprime polynomials, we establish some $q$-supercongruences modulo the third power of a cyclotomic polynomial. In particular, we give a $q$-analogue of a formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808].