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Decimal expansion of the odd Bessel moment t(5,3) (see the referenced paper about the elliptic integral evaluations of Bessel moments).
2

%I #6 Jun 06 2016 11:24:17

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

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

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

%N Decimal expansion of the odd Bessel moment t(5,3) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

%H David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, <a href="http://arxiv.org/abs/0801.0891">Elliptic integral evaluations of Bessel moments</a>, arXiv:0801.0891 [hep-th], 2008, page 21.

%F Integral_{0..inf} x*BesselI_0(x)^2*BesselK_0(x)^3.

%e 0.252253944897841965994509612555090408775068450755970099920659309452897...

%t t[5, 3] = NIntegrate[x^3*BesselI[0, x]^2*BesselK[0, x]^3, {x, 0, Infinity}, WorkingPrecision -> 104];

%t RealDigits[t[5, 3]][[1]]

%t (* or: *)

%t K[k_] := EllipticK[k^2/(-1+k^2)]/Sqrt[1-k^2];

%t D0[y_] := (4*y*K[Sqrt[((1-3*y)*(1+y)^3)/((1+3*y)*(1-y)^3)]])/Sqrt[(1+3*y)* (1-y)^3];

%t t[5, 3] = NIntegrate[4y^2*(1-2y^2+4y^4)*D0[y]/(1-4y^2)^(5/2), {y, 0, 1/3}, WorkingPrecision -> 104];

%t RealDigits[t[5, 3]][[1]]

%Y Cf. A073010 (s(3,1)), A121839 (1+s(3,3)), A222068 (s(4,1)), A244854 (2s(4,3)), A273959, A273984 (s(5,1)), A273985 (s(5,3)), A273986 (s(5,5)), A273989 (t(5,1)), A273991 (t(5,5)).

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Jun 06 2016