A257437 - OEIS

%I #19 Apr 05 2024 11:10:14

%S 9,1,2,0,5,5,0,8,9,3,7,2,7,1,8,4,0,0,0,1,8,8,2,8,6,5,5,6,9,3,0,9,7,5,

%T 3,4,4,5,1,1,4,2,9,1,1,9,8,0,3,1,8,7,4,6,8,4,3,6,2,4,2,6,1,7,8,4,7,8,

%U 3,6,9,1,3,8,3,6,5,2,7,4,1,1,1,5,0,3,4,6,4,5,0,4,7,5,7,7,3,5,5,7,2,6,5

%N Decimal expansion of G(1/12), a generalized Catalan constant.

%H G. C. Greubel, <a href="/A257437/b257437.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(1/12) = 3*(sqrt(3)+1)*(log(sqrt(2)-1) + (sqrt(3)/2)*log(sqrt(3)+sqrt(2))).

%e 0.91205508937271840001882865569309753445114291198031874684362426...

%t RealDigits[3*(Sqrt[3]+1)*(Log[Sqrt[2]-1] + (Sqrt[3]/2)*Log[Sqrt[3]+ Sqrt[2]]), 10, 103] // First

%t N[Pi*HypergeometricPFQ[{5/12, 1/2, 7/12}, {1, 3/2}, 1]/4, 106] (* _Vaclav Kotesovec_, Apr 24 2015 *)

%o (PARI) 3*(sqrt(3)+1)*(log(sqrt(2)-1) + (sqrt(3)/2)*log(sqrt(3)+ sqrt(2))) \\ _G. C. Greubel_, Aug 24 2018

%o (Magma) SetDefaultRealField(RealField(100)); 3*(Sqrt(3)+1)*(Log(Sqrt(2)-1) + (Sqrt(3)/2)*Log(Sqrt(3)+ Sqrt(2))) // _G. C. Greubel_, Aug 24 2018

%Y Cf. A006752 (G(0) = Catalan), A257435 (G(1/6)), A091648 (G(1/4)), A257436 (G(1/3)), A257438 (G(1/5)).

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Apr 23 2015