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$q$-Analogues of two Ramanujan-type formulas for $1/π$ (1802.01944v2)

Published 6 Feb 2018 in math.NT and math.CO

Abstract: We give $q$-analogues of the following two Ramanujan-type formulas for $1/\pi$: \begin{align*} \sum_{k=0}\infty (6k+1)\frac{(\frac{1}{2})k3}{k!3 4k} =\frac{4}{\pi} \quad\text{and}\quad \sum{k=0}\infty (-1)k(6k+1)\frac{(\frac{1}{2})_k3}{k!3 8k } =\frac{2\sqrt{2}}{\pi}. \end{align*} Our proof is based on two $q$-WZ pairs found by the first author in his earlier work.

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