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 \(\frac{1}{\pi^2}\).


33C20 Generalized hypergeometric series, \({}_pF_q\)
33F05 Numerical approximation and evaluation of special functions
11Y55 Calculation of integer sequences
Full Text: DOI arXiv


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