A257435 - OEIS

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

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

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

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

%N Decimal expansion of G(1/6), a generalized Catalan constant.

%H G. C. Greubel, <a href="/A257435/b257435.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(1/6) = (3/4)*sqrt(3)*log(2).

%e 0.900424600389707757858827589029049485829943957666618765595157318391...

%t RealDigits[(3/4)*Sqrt[3]*Log[2], 10, 105] // First

%t N[Pi*HypergeometricPFQ[{1/3, 1/2, 2/3}, {1, 3/2}, 1]/4, 105] (* _Vaclav Kotesovec_, Apr 24 2015 *)

%o (PARI) (3/4)*sqrt(3)*log(2) \\ _G. C. Greubel_, Aug 24 2018

%o (Magma) SetDefaultRealField(RealField(100)); (3/4)*Sqrt(3)*Log(2); // _G. C. Greubel_, Aug 24 2018

%Y Cf. A006752 (G(0) = Catalan), A091648 (G(1/4)), A257436 (G(1/3)), A257437 (G(1/12)), A257438 (G(1/5)).

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Apr 23 2015