A $q$-ANALOGUE OF A HYPERGEOMETRIC CONGRUENCE. Issue 2 (18th April 2020) - GU, CHENG-YANG; GUO, VICTOR J. W.
Abstract : We give a $q$ -analogue of the following congruence: for any odd prime $p$, $$\begin{eqnarray}\mathop{\sum }_{k=0}^{(p-1)/2}(-1)^{k}(6k+1)\frac{(\frac{1}{2})_{k}^{3}}{k!^{3}8^{k}}\mathop{\sum }_{j=1}^{k}\biggl(\frac{1}{(2j-1)^{2}}-\frac{1}{16j^{2}}\biggr)\equiv 0\;(\text{mod}\;p), \end{eqnarray}$$ which was originally conjectured by Long and later proved by Swisher. This confirms a conjecture of the second author ['A $q$ -analogue of the (L.2) supercongruence of Van Hamme', J. Math. Anal. Appl.  466 (2018), 749–761].