Some q-supercongruences related to Swisher’s (H.3) conjecture

Abstract

We first give a [Formula: see text]-analogue of a supercongruence of Sun, which is a generalization of Van Hamme’s (H.2) supercongruence for any prime [Formula: see text]. We also give a further generalization of this [Formula: see text]-supercongruence, which may also be considered as a generalization of a [Formula: see text]-supercongruence recently conjectured by the second author and Zudilin. Then, by combining these two [Formula: see text]-supercongruences, we obtain [Formula: see text]-analogues of the following two results: for any integer [Formula: see text] and prime [Formula: see text] with [Formula: see text] [Formula: see text] [Formula: see text] which are generalizations of Swisher’s (H.3) conjecture modulo [Formula: see text] for [Formula: see text]. The key ingredients in our proof are the ‘creative microscoping’ method, the [Formula: see text]-Dixon sum, Watson’s terminating [Formula: see text] transformation, and properties of the [Formula: see text]-adic Gamma function.