q-Analogues of Dwork-type supercongruences

2019/10/10 by Guo, Victor J. W.
#11A07 #11B65 #33D15 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1910.07551

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].

Related

• q-analogues of the G.2 Supercongruence of Van Hamme
• Further q-analogues of the (G.2) Supercongruence of Van Hamme
• Refinements of Van Hamme's (E.2) and (F.2) supercongruences and two supercongruences by Swisher
• A -ANALOGUE OF A DWORK-TYPE SUPERCONGRUENCE
• A refined q-analogue of some congruences of Van Hamme