Hello, the formula for (1/4)! is quite interesting, the approximation is 88 digits, this is unique. Now about Gamma(n/48), I do not think that any higher value would lead to simple approximations as mr Gosper showed, as the index increases : it gets quite messy, we can only hope to get the first denominators in my opinion. Nevertheless, the values for (1/3)! and (1/4)! are impressive, the technique uses <approximations> of dedekind functions, a neat trick. best regards, Simon Plouffe Le 28/12/2011 18:43, Joerg Arndt a écrit :
* Warren Smith<warren.wds@gmail.com> [Dec 28. 2011 18:30]:
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Then I tried Gosper's (1/4)! and I don't understand how Gosper got his formula. Actually, I practically never understand how Gosper gets his formulas. As I mentioned before, it'd be nice to find formulas for GAMMA(k/48) for example. Such formulas might exist in terms of algebraic numbers and the AGM.
Just as quick copy and paste (suggest to start with the last one):
J.\ M.\ Borwein, I.\ J.\ Zucker: {Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind}, IMA Journal of Numerical Analysis, vol.12, no.4, pp.519-526, \bdate{1992}.}
Greg Martin: {A product of Gamma function values at fractions with the same denominator}, arXiv:0907.4384v1 [math.CA], \bdate{24-July-2009}. URL: \url{http://arxiv.org/abs/0907.4384}.}
Albert Nijenhuis: {Small Gamma Products with Simple Values}, arXiv:0907.1689v1 [math.CA], \bdate{9-July-2009}. URL: \url{http://arxiv.org/abs/0907.1689}.}
Raimundas Vid\={u}nas: {Expressions for values of the gamma function}, arXiv:math.CA/0403510, \bdate{30-March-2004}. URL: \url{http://arxiv.org/abs/math/0403510}.}
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