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

Abstract

Using Watson's terminating ${}_8{\varphi }_7$ 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.

6 pages

References in corpus (2)

• Some new $q$-congruences for truncated basic hypergeometric series: even powers
• On a $q$-deformation of modular forms

Cited by in corpus (1)

• On a conjectural congruence of Guo