## $$q$$-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.

### MSC:

 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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### References:

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