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

We first give a q -analogue of a supercongruence of Sun, which is a generalization of Van Hamme's (H.2) supercongruence for any prime p ≡ 3 (mod 4). We also give a further generalization of this q -supercongruence, which may also be considered as a generalization of a q -supercongruence recently conjectured by the second author and Zudilin. Then, by combining these two q -supercongruences, we obtain q -analogues of the following two results: for any integer d > 1 and prime p with p ≡ − 1 (mod 2 d) ∑ k = 0 (p 2 − 1) / d 1 d k 3 k ! 3 ≡ p 2 (mod p 4) , ∑ k = 0 p 2 − 1 1 d k 3 k ! 3 ≡ p 2 (mod p 4) , which are generalizations of Swisher's (H.3) conjecture modulo p 4 for r = 2. The key ingredients in our proof are the 'creative microscoping' method, the q -Dixon sum, Watson's terminating 8 ϕ 7 transformation, and properties of the p -adic Gamma function. [ABSTRACT FROM AUTHOR]