A273990 Decimal expansion of the odd Bessel moment t(5,3) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

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, 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, 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

Offset

0,1

Links

Table of n, a(n) for n=0..103.

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008, page 21.

Formula

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

Example

0.252253944897841965994509612555090408775068450755970099920659309452897...

Mathematica

t[5, 3] = NIntegrate[x^3*BesselI[0, x]^2*BesselK[0, x]^3, {x, 0, Infinity}, WorkingPrecision -> 104];
RealDigits[t[5, 3]][[1]]
(* or: *)
K[k_] := EllipticK[k^2/(-1+k^2)]/Sqrt[1-k^2];
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[5, 3] = NIntegrate[4y^2*(1-2y^2+4y^4)*D0[y]/(1-4y^2)^(5/2), {y, 0, 1/3}, WorkingPrecision -> 104];
RealDigits[t[5, 3]][[1]]

Crossrefs

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)).

Sequence in context: A171937 A065291 A065267 * A216022 A100955 A049413

Adjacent sequences: A273987 A273988 A273989 * A273991 A273992 A273993

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 06 2016

Status

approved