Some variations of a ‘divergent’ Ramanujan-type<i>q</i>-supercongruence

Abstract

Using the q-Wilf–Zeilberger method and a q-analogue of a ‘divergent’ Ramanujan-type supercongruence, we give several q-supercongruences modulo the fourth power of a cyclotomic polynomial. One of them is a q-analogue of a supercongruence recently proved by Wang: for any prime p>3, ∑k=0p−1(3k−1)(12)k(−12)k2k!34k≡p−2p3(modp4),where (a)k=a(a+1)⋯(a+k−1) is the Pochhammer symbol.

Topics & Concepts