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%I #18 Apr 05 2024 11:10:14
%S 8,9,3,6,7,1,4,2,3,4,6,0,9,6,3,5,5,4,3,0,2,0,6,9,8,5,4,5,8,3,5,4,6,0,
%T 0,7,5,4,7,5,5,8,0,9,4,7,9,6,3,2,8,0,7,8,2,2,0,3,0,8,5,8,4,8,7,8,1,5,
%U 7,6,4,1,7,7,0,4,9,2,9,1,5,0,7,9,6,7,0,5,1,6,3,8,4,2,2,3,7,2,8,1,4,8,0,3
%N Decimal expansion of G(1/5), a generalized Catalan constant.
%H G. C. Greubel, <a href="/A257438/b257438.txt">Table of n, a(n) for n = 0..10000</a>
%H D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, <a href="https://carmamaths.org/resources/jon/emoments.pdf">Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant</a>, 2010.
%H D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, <a href="https://doi.org/10.1016/j.jmaa.2011.06.001">Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant</a>, Journal of Mathematical Analysis and Applications, Volume 384, Issue 2, 15 December 2011, Pages 478-496.
%F G(s) = (Pi/4) * 3F2(1/2, 1/2-s, s+1/2; 1, 3/2; 1), with 2F1 the hypergeometric function.
%F 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).
%F 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)))).
%e 0.8936714234609635543020698545835460075475580947963280782203...
%t 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
%t N[Pi*HypergeometricPFQ[{3/10, 1/2, 7/10}, {1, 3/2}, 1]/4, 105] (* _Vaclav Kotesovec_, Apr 24 2015 *)
%o (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
%o (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
%Y Cf. A006752 (G(0) = Catalan), A257435 (G(1/6)), A091648 (G(1/4)), A257436 (G(1/3)), A257437 (G(1/12)).
%K nonn,cons,easy
%O 0,1
%A _Jean-François Alcover_, Apr 23 2015