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%I #14 Oct 01 2018 04:10:45
%S 1,8,7,4,4,7,1,6,6,9,4,9,0,0,8,2,6,0,1,1,8,0,9,5,0,9,9,9,4,8,9,6,8,0,
%T 2,9,7,0,5,7,3,9,7,6,5,8,9,2,0,3,7,9,5,3,4,8,0,7,6,9,8,4,5,1,1,9,0,4,
%U 5,2,6,4,7,5,6,8,0,0,7,0,0,3,7,5,8,4,7,0,6,5,3,3,9,9,9,8,9,8,0,4,3
%N Decimal expansion of Q(4), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).
%H David J. Broadhurst, <a href="http://arxiv.org/abs/hep-th/9803091">Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity</a>, arXiv:hep-th/9803091, 1998, p. 12.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/ClausensIntegral.html">Clausen's Integral</a>
%F Q(n) = Integral_{0..inf} arccosh((x+2)/2)^2 log((x+1)/x)/(x+n) dx.
%F Computation is done using the analytical form given by David Broadhurst:
%F Q(4) = 125/54 zeta(4) + 8 U - 8 V, where U is A255685 and V A274400.
%e 1.87447166949008260118095099948968029705739765892037953480769845119...
%t digits = 101;
%t U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];
%t v[k_] := ((-1)^k*((24*(k - 1)*(3*k - 4))/(3*k - 2)^3 + (8*(3*k*(3*k - 5) + 4))/(27*(k - 1)^3) + PolyGamma[2, (3*k)/2 - 1] - PolyGamma[2, (3*(k - 1))/2]))/(48*(k - 1)*(3*k - 4)*(3*k - 2));
%t V = A274400 = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"];
%t Q[4] = 125/54 Zeta[4] + 8 U - 8 V;
%t RealDigits[Q[4], 10, digits][[1]]
%o (PARI)
%o Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
%o Q(4) \\ _Gheorghe Coserea_, Sep 30 2018
%o (PARI)
%o polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
%o u31=Pi^4/180 + (Pi^2/12)*log(2)^2 - (1/12)*log(2)^4 - 2*polylog(4, 1/2);
%o v31=3*zeta(3)/8 - 1/2 + sumalt(k=2, (-1)^k*((24*(k-1)*(3*k-4))/(3*k-2)^3 + (8*(3*k*(3*k-5)+4))/(27*(k-1)^3) + polygamma(2, (3*k)/2-1) - polygamma(2, (3*(k-1))/2))/(48*(k-1)*(3*k-4)*(3*k-2)));
%o 125/54*zeta(4) + 8*u31 - 8*v31 \\ _Gheorghe Coserea_, Sep 30 2018
%Y Cf. A274438 (Q(0)), A274439 (Q(1)), A274440 (Q(2)), A274441 (Q(3)).
%K nonn,cons
%O 1,2
%A _Jean-François Alcover_, Jun 23 2016
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