$q$-Analogues of Dwork-type supercongruences

TL;DR

This paper proves new $q$-analogues of Dwork-type supercongruences related to Van Hamme's conjectures, using the creative microscoping method, and raises related conjectures on $q$-analogues.

Contribution

It establishes $q$-analogues of specific supercongruences, improving the known modulus bounds and introducing the creative microscoping method for such proofs.

Findings

Proved $q$-analogues of (C.3) and (J.3) supercongruences modulo $p^{3r}$.

Established new conjectures on $q$-analogues of Van Hamme's supercongruences.

Applied the creative microscoping method in the proof process.

Abstract

In 1997, Van Hamme conjectured 13 Ramanujan-type supercongruences. All of the 13 supercongruences have been confirmed by using a wide range of methods. In 2015, Swisher conjectured Dwork-type supercongruences related to the first 12 supercongruences of Van Hamme. Here we prove that the (C.3) and (J.3) supercongruences of Swisher are true modulo $p^{3r}$ (the original modulus is $p^{4r}$) by establishing $q$-analogues of them. Our proof will use the creative microscoping method, recently introduced by the author in collaboration with Zudilin. We also raise conjectures on $q$-analogues of an equivalent form of the (M.2) supercongruence of Van Hamme, partially answering a question at the end of [Adv. Math. 346 (2019), 329--358].