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arXiv · 1802.01944

$q$-Analogues of two Ramanujan-type formulas for $1/\pi$

Victor J. W. Guo·Ji-Cai Liu
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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})_k^3}{k!^3 4^k} =\frac{4}{\pi} \quad\text{and}\quad \sum_{k=0}^\infty (-1)^k(6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 8^k } =\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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Victor J. W. Guo, Ji-Cai Liu. 2018-02-06. $q$-Analogues of two Ramanujan-type formulas for $1/\pi$. https://arxiv.org/abs/1802.01944

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