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A \(q\)-analogue of a Ramanujan-type supercongruence involving central binomial coefficients. (English) Zbl 1373.05025

Summary: Motivated by W. Zudilin’s work [J. Number Theory 129, No. 8, 1848–1857 (2009; Zbl 1231.11147)], we give a \(q\)-analogue of a Ramanujan-type supercongruence of L. van Hamme [Lect. Notes Pure Appl. Math. 192, 223–236 (1997; Zbl 0895.11051)] and E. Mortenson [Proc. Am. Math. Soc. 136, No. 12, 4321–4328 (2008; Zbl 1171.11061)] via the \(q\)-WZ method. Meanwhile, we give a \(q\)-analogue of a related congruence of Z.-W. Sun [Electron. J. Comb. 20, No. 1, Research Paper P9, 14 p. (2013; Zbl 1266.05004)] in the same way. We also propose several related conjectures on congruences involving central \(q\)-binomial coefficients.

MSC:

05A30 \(q\)-calculus and related topics
05A10 Factorials, binomial coefficients, combinatorial functions
11B65 Binomial coefficients; factorials; \(q\)-identities
11A07 Congruences; primitive roots; residue systems
11F33 Congruences for modular and \(p\)-adic modular forms
11Y55 Calculation of integer sequences
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