From: Raymond Manzoni <raymman@club-internet.fr>
Subject: Re: Gamma(1/4)
Date: Sat, 01 May 1999 21:02:26 +0200
Newsgroups: sci.math

Felipe Voloch wrote:

> G. A. Edgar (edgar@math.ohio-state.edu.nospam) wrote:
> : In article <7gcdb7$om7$1@nnrp1.dejanews.com>, Sir-G
> : <sirg@my-dejanews.com> wrote:
> :
> : >  Hello, everybody!
> : >  Does anybody know about the following question?
> : >  Can Gamma(1/4) be calculated (e.g. in terms of pi)?
> : >  Is it irrational (transcedental)?
> : > P.S. Gamma here is Euler's Gamma function. (Gamma(n)=(n-1)!).
> : > --
> :
> : I think nothing is known like that.  You can say
> : Gamma(1/4)*Gamma(3/4) = pi*sqrt(2), but nothing for either one
> : separately.
>
> pi, e^pi and Gamma(1/4) are algebraically independent!
> This means that (to answer the original poster's question)
> that Gamma(1/4) is indeed transcendental but can't be
> computed in terms of pi. This was proved a few years ago by Nesterenko.
>
> Felipe

    Hello,

    First of all I must admit I was involved in exactly the same quest
some time ago.

    If you want to know more (concerning this interesting reference to
Nesterenko or more generally concerning Gamma function) look at the very
interesting :

    Steve Finch's page (and M. Waldschmidt and F. Gramain references) at
        http://www.mathsoft.com/asolve/constant/gamma/gamma.html
    or Eric Weisstein page at :
       http://www.astro.virginia.edu/~eww6n/math/GammaFunction.html

    or go to Yuri Nesterenko's home page at :
       http://www.math.jussieu.fr/~nesteren/

    I hope all that will give you an accurate idea of actual state of the
art.
    With my best wishes,

                    Raymond Manzoni