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A \(q\)-analogue of the (L.2) supercongruence of van Hamme. (English) Zbl 1405.33021

Summary: The (L.2) supercongruence of van Hamme was proved by Swisher recently. In this paper we provide a conjectural \(q\)-analogue of the (L.2) supercongruence of van Hamme and prove a weaker form of it by using the \(q\)-WZ method. In the same way, we prove a complete \(q\)-analogue of the following congruence \[ \mathop{\sum}\limits_{k = 0}^n(6 k + 1)\begin{pmatrix} 2 k \\ k \end{pmatrix}^3(- 512)^{n - k} \equiv 0\quad (\operatorname{mod} 4(2 n + 1)\begin{pmatrix} 2 n \\ n \end{pmatrix}), \] which was conjectured by Z. -W. Sun and confirmed by B. He. We also provide a conjectural \(q\)-analogue of another congruence proved by Swisher.

MSC:

33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
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