OFFSET
0,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant, 2010.
D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant, Journal of Mathematical Analysis and Applications, Volume 384, Issue 2, 15 December 2011, Pages 478-496.
FORMULA
G(s) = (Pi/4) * 3F2(1/2, 1/2-s, s+1/2; 1, 3/2; 1), with 2F1 the hypergeometric function.
G(s) = (1/(8*s))*(Pi + cos(Pi*s)*(psi(1/4+s/2) - psi(3/4+s/2))), where psi is the digamma function (PolyGamma).
G(1/5) = (5/8)*sqrt(5+2*sqrt(5))*(((sqrt(5)-1)/2)*arcsinh(sqrt(5+2*sqrt(5))) - arcsinh(sqrt(5-2*sqrt(5)))).
EXAMPLE
0.8936714234609635543020698545835460075475580947963280782203...
MATHEMATICA
RealDigits[(5/8)*Sqrt[5+2*Sqrt[5]]*(((Sqrt[5]-1)/2)*ArcSinh[Sqrt[5+2*Sqrt[5]]] - ArcSinh[Sqrt[5-2*Sqrt[5]]]), 10, 104] // First
N[Pi*HypergeometricPFQ[{3/10, 1/2, 7/10}, {1, 3/2}, 1]/4, 105] (* Vaclav Kotesovec, Apr 24 2015 *)
PROG
(PARI) (5/8)*sqrt(5+2*sqrt(5))*(((sqrt(5)-1)/2)*asinh(sqrt(5 +2*sqrt(5))) - asinh(sqrt(5-2*sqrt(5)))) \\ G. C. Greubel, Aug 24 2018
(Magma) SetDefaultRealField(RealField(100)); (5/8)*Sqrt(5+2*Sqrt(5))*(((Sqrt(5)-1)/2)*Argsinh(Sqrt(5+2*Sqrt(5))) - Argsinh(Sqrt(5-2*Sqrt(5)))); // G. C. Greubel, Aug 24 2018
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Apr 23 2015
STATUS
approved