Cheng-Yang Gu

In this note we use some $q$ -congruences proved by the method of ‘creative microscoping’ to prove two conjectures on supercongruences involving central binomial coefficients. In particular, we confirm the $m=5$ case of Conjecture 1.1 of Guo [‘Some generalizations of a supercongruence of Van Hamme’, Integral Transforms Spec. Funct.28 (2017), 888–899].

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].