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  • CHENG-YANG GU (a1) and VICTOR J. W. GUO (a2)


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].


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The second author was partially supported by the National Natural Science Foundation of China (grant 11771175).



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[12] Straub, A., ‘Supercongruences for polynomial analogs of the Apéry numbers’, Proc. Amer. Math. Soc. 147 (2019), 10231036.
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[14] Van Hamme, L., ‘Some conjectures concerning partial sums of generalized hypergeometric series’, in: p-Adic Functional Analysis (Nijmegen, 1996), Lecture Notes in Pure and Applied Mathematics, 192 (Dekker, New York, 1997), 223236.
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