New q-Analogues of Van Hamme's (E.2) Supercongruence and of a Supercongruence by Swisher

Abstract

In this paper, a couple of q-supercongruences for truncated basic hypergeometric series are proved, most of them modulo the cube of a cyclotomic polynomial. One of these results is a new q-analogue of the (E.2) supercongruence by Van Hamme, another one is a new q-analogue of a supercongruence by Swisher, while the other results are closely related q-supercongruences. The proofs make use of special cases of a very-well-poised ${}_6{\varphi }_5$ summation. In addition, the proofs utilize the method of creative microscoping (which is a method recently introduced by the first author in collaboration with Wadim Zudilin), and the Chinese remainder theorem for coprime polynomials.

Keywords: 6 ϕ 5 summation; Basic hypergeometric series; cyclotomic polynomial; q-congruences; supercongruences.