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A340826 Decimal expansion of Cl_2(Pi/5), where Cl_2 is the Clausen function of order 2.

%I A340826 #26 Jun 11 2025 02:53:34
%S A340826 9,2,3,7,5,5,1,6,8,1,0,0,5,3,5,3,0,8,7,1,1,9,8,6,0,2,9,7,9,3,0,2,4,3,
%T A340826 5,3,9,6,6,2,7,9,0,0,6,4,1,2,5,1,7,2,5,1,7,0,7,7,1,2,8,4,8,3,2,5,1,5,
%U A340826 0,9,8,3,3,2,5,3,0,9,7,5,7,2,8,7,2,8,3,2,2,1,8,0,1,1,2,2,5,9,9,9,6,2,6,3,5
%N A340826 Decimal expansion of Cl_2(Pi/5), where Cl_2 is the Clausen function of order 2.
%H A340826 For links see A261024.
%F A340826 A = Cl_2(Pi/5).
%F A340826 B = Cl_2(2*Pi/5).
%F A340826 C = Cl_2(3*Pi/5).
%F A340826 D = Cl_2(4*Pi/5).
%F A340826 4*(A^2 + C^2) = 5*(B^2 + D^2).
%F A340826 B = 2*A - 2*D.
%F A340826 D = 2*B - 2*C.
%F A340826 2*C = 4*A - 5*D.
%F A340826 B = -D + sqrt(A*(2*C+D)+D^2).
%F A340826 B^2 + D^2 = 4*Pi^4/(325*A340628^2).
%F A340826 B^2 + D^2 = (13/1125)*A340629^2*Pi^4.
%F A340826 Equals Pi*(2*log(G(9/10) / G(11/10)) + log(Pi*(1+sqrt(5)))/5), where G is the Barnes G-function. - _Vaclav Kotesovec_, Jan 23 2021
%e A340826 0.9237551681005353087119860297930...
%t A340826 Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Re[Cl2[Pi/5]], 10, 105] // First
%t A340826 N[Pi*(ArcCsch[2] + Log[2*Pi*BarnesG[9/10]^10 / BarnesG[11/10]^10])/5, 120] (* _Vaclav Kotesovec_, Jan 23 2021 *)
%Y A340826 Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261025 (Cl_2(Pi/4)), A261026 (Cl_2(3*Pi/4)), A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)), A340628, A340629.
%K A340826 nonn,cons
%O A340826 0,1
%A A340826 _Artur Jasinski_, Jan 23 2021