$q$-Supercongruences from Jackson's ${}_8{\varphi }_7$ summation and Watson's ${}_8{\varphi }_7$ transformation

Abstract

$q$-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some $q$-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's ${}_8{\varphi }_7$ summation, Watson's ${}_8{\varphi }_7$ transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials. More concretely, we give a $q$-analogue of a nice formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808] and two $q$-supercongruences involving double series.

References in corpus (3)

• Congruences on sums of $q$-binomial coefficients
• Dwork-type supercongruences through a creative $q$-microscope
• Two $q$-supercongruences from Watson's transformation