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Quadratic transformations and Guillera’s formulas for $$1/\pi^2$$. (English. Russian original) Zbl 1144.33002
Math. Notes 81, No. 3, 297-301 (2007); translation from Mat. Zametki 81, No. 3, 335-340 (2007).
The quadratic transformations for $${}_{2}F_{1}$$ series, ${}_{2}F_{1} (a,b; 1+a-b; z) = (1-z)^{-a} \cdot \;{}_{2}F_{1} \left(\frac{1}{2} a, \frac{1}{2} + \frac{1}{2} a-b; 1+a-b; \frac{-4z}{(1-z)^2}\right)$ due to Gauss, and for $${}_{3}F_{2}$$ series, $\begin{split}{}_{3}F_{2} (a,b,c; 1+a-b, 1+a-c; z) \\= (1-z)^{-a} \cdot {}_{3}F_{2} \left(\frac{1}{2} a, \frac{1}{2} + \frac{1}{2} a, 1+a-b-c; 1+a-b, 1+a-c; \frac{-4z}{(1-z)^2}\right)\end{split}$ due to Whipple are well-known.
In this paper, the author proves an analogue for $${}_{5}F_{4}$$ series that involves multiple hypergeometric series. By utilizing this quadratic transformation formula and results of J. Guillera, formulas for $$\frac{1}{\pi^2}$$ which are given by certain $${}_{5}F_{4}$$ hypergeometric series, he derives two new formulas for $$\frac{1}{\pi^2}.$$
This idea seems likely to be used to derive more formulas for $$\frac{1}{\pi^2}$$.

##### MSC:
 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33F05 Numerical approximation and evaluation of special functions 11Y55 Calculation of integer sequences
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##### References:
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