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Boris Gourévitch
The world of Pi - V2.57
modif. 13/04/2013



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14 Series of greater factorial : Ramanujan, Borwein, Chudnovsky....

14.1 Squared central binomial coefficients: Elliptical formulae

This case quickly gets complicated and it's only to show a glimpse of it that I talk about it here, since in fact, we loose the family of constant we're studying to land on some elliptical integrals... It's due to the fact that

 sum  oo  ( k)2(x-)2k  2-
    C 2k    4   =  pK (x)
k=0
(713)

We are in all cases stuck, the values are not simple constant of a certain order anymore, but appear under the form of products and factors of the Gamma functions. It seems that this is the case for the powers of the combinations, more generaly, as the next section shows it.

14.2 Cubic central binomial coefficients: Ramanujan's Identity

It gets even more complicated... but we are still staying in the elliptical...

pict


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