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arxiv: 1802.01944 · v2 · pith:GD2EZZ7Xnew · submitted 2018-02-06 · 🧮 math.NT · math.CO

q-Analogues of two Ramanujan-type formulas for 1/π

classification 🧮 math.NT math.CO
keywords fracalignanaloguesformulasinftyquadramanujan-typeauthor
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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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