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A new method to obtain series for \(1/\pi\) and \(1/\pi^2\). (English) Zbl 1113.11076

The author conjectures several Ramanujan-type series for \(1/\pi\) and for \(1/\pi^2\). As evidence supporting their validity he presents several examples that have been checked numerically with a precision of 200 digits. For example, \[ \sum_ {n=0}^ \infty\sum_ {j=0}^ n "\begin{pmatrix} 2j\\ j\end{pmatrix}^ 2 \begin{pmatrix} 2n-2j\\ n-j\end{pmatrix}^ 2\left(\frac{2-\sqrt{3}}{64}\right)^ n \left(\frac{1}{4}+\frac{3+2\sqrt{3}}{4}n\right)=\frac{1}{\pi}. \]

MSC:

11Y60 Evaluation of number-theoretic constants
33C20 Generalized hypergeometric series, \({}_pF_q\)
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Online Encyclopedia of Integer Sequences:

Decimal expansion of 1/Pi.