\(q\)-analogues of two Ramanujan-type formulas for \(1/\pi\). (English) Zbl 1444.11036

Summary: We give \(q\)-analogues of the following two Ramanujan-type formulas for \(1/\pi\): \[ \sum^\infty_{k=0} (6k+1) \frac{(\frac{1}{2})^3_k}{k!^3 4^k}=\frac{4}{\pi}\quad\text{ and }\quad\sum^\infty_{k=0}(-1)^k(6k+1)\frac{(\frac{1}{2})^3_k}{k!^3 8^k}=\frac{2\sqrt{2}}{\pi}. \] Our proof is based on two \(q\)-WZ pairs found by the first author in his earlier work.


11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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