Some<i>q</i>-congruences on double basic hypergeometric sums

Abstract

We give three q-congruences on double basic hypergeometric sums. One of them is a q-analogue of the following supercongruence: for any prime p>3, ∑k=0(p−1)/2(4k+1)(12)k4k!4∑j=1k(1(2j−1)2−14j2)≡0(modp2). Our proof uses q-analogues of two Ramanujan-type supercongruences of Van Hamme and a q-analogue of a ‘divergent’ Ramanujan-type supercongruence.

Topics & Concepts