q-Analogues of Dwork-type supercongruences

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].

Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

“…We refer the reader to [1,7,[16][17][18][19][20][21][22][23][24]27,36,39,45,49,53,54,57,58] for some interesting q-congruences.…”

A family of <inline-formula><tex-math id="M1">$ q $</tex-math></inline-formula>-congruences modulo the square of a cyclotomic polynomial

“…The m = 1 case of (1.3) was first conjectured by the author and Zudilin [9, Conjecture 4.13] and has already been proved by themselves in a recent paper [11]. For some other recent progress on q-congruences, the reader may consult [2,3,4,5,6,7,8,10,15].…”

Three supercongruences for Apery numbers and Franel numbers

“…It is easy to see that (1.3) and (1.4) are congruences of this type. We refer the reader to [2]- [4] for Dwork-type congruences, and to [9], [11], [13] and [19] for q-analogues of Dwork-type congruences.…”