Some $q$-supercongruences modulo the square and cube of a cyclotomic polynomial

TL;DR

This paper proves new $q$-supercongruences involving cyclotomic polynomials, extending previous conjectures and employing hypergeometric series identities to establish congruences modulo higher powers.

Contribution

It introduces novel $q$-supercongruences with parameters, extending earlier conjectures and demonstrating congruences modulo the square, cube, and possibly the fourth power of cyclotomic polynomials.

Findings

Established $q$-supercongruences modulo the square and cube of cyclotomic polynomials.

Partly confirmed earlier conjectures by the authors.

Proposed conjecture for congruence modulo the fourth power of a cyclotomic polynomial.

Abstract

Two $q$-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two $q$-supercongruences that were earlier conjectured by the same authors and involve $q$-supercongruences modulo the square and the cube of a cyclotomic polynomial. One of the newly proved $q$-supercongruences is even conjectured to hold modulo the fourth power of a cyclotomic polynomial.