New q-analogues of Van Hamme's (E.2) supercongruence
and of a supercongruence by Swisher
(13 pages)
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
6φ5 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.
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