## 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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### References:

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