A family of $q$-congruences modulo the square of a cyclotomic polynomial

TL;DR

This paper proves a family of $q$-congruences involving cyclotomic polynomials using Watson's transformation, confirming conjectures and deriving supercongruences and $q$-analogues, advancing the understanding of $q$-series and supercongruences.

Contribution

It introduces a new family of $q$-congruences modulo squared cyclotomic polynomials, confirming conjectures and deriving related supercongruences and $q$-analogues.

Findings

Proved a family of $q$-congruences modulo squared cyclotomic polynomials.

Derived supercongruences modulo $p^4$ and their $q$-analogues.

Partially confirmed a case of Swisher's conjecture.

Abstract

Using Watson's terminating $_8\phi_7$ transformation formula, we prove a family of $q$-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636--646]. As an application, we deduce two supercongruences modulo $p^4$ ($p$ is an odd prime) and their $q$-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.