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\(\pi\) and some other constants. (English) Zbl 1127.33003

The author considers a particular integral and reduces it to hypergeometric form. Using this result, some numerical constants including \(\pi\) are represented as Gauss hypergeometric series. These series have half integer parameters, for example \(_2F_1(7/2,7/2;9/2;1/2)\) (Example 3.3) and can therefore be evaluated (even at a generic point like \(_2F_1(7/2,7/2;8/2;x)\)).
In principle such evaluations of Gauss hypergeometric series can be handled by an algorithm of Kelly Roach [Proc. ISSAC 1996, Zürich, 301–308 (1996; Zbl 0914.65009)].

MSC:

33C05 Classical hypergeometric functions, \({}_2F_1\)
33B15 Gamma, beta and polygamma functions
11Y60 Evaluation of number-theoretic constants

Citations:

Zbl 0914.65009
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