A273989 Decimal expansion of the odd Bessel moment t(5,1) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

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.

Equals 4(7 - 4*sqrt(3)) EllipticK(1 - 32/(16 + 7*sqrt(3) - sqrt(15))) EllipticK(1 - 32/(16 + 7*sqrt(3) + sqrt(15))).

Example

0.660344869018672357837266831705994263854241991696873858300803587553894...

Mathematica

t[5, 1] = NIntegrate[x*BesselI[0, x]^2*BesselK[0, x]^3, {x, 0, Infinity}, WorkingPrecision -> 105]; RealDigits[t[5, 1]][[1]]
(* or: *)
t[5, 1] = 4(7 - 4*Sqrt[3]) EllipticK[1 - 32/(16 + 7*Sqrt[3] - Sqrt[15])] EllipticK[1 - 32/(16 + 7*Sqrt[3] + Sqrt[15])]; RealDigits[t[5, 1], 10, 105][[1]]
RealDigits[EllipticK[(16 - 7 Sqrt[3] - Sqrt[15])/32] EllipticK[(16 - 7 Sqrt[3] + Sqrt[15])/32]/4, 10, 105][[1]] (* Jan Mangaldan, Jan 06 2017 *)

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)), A273990 (t(5,3)), A273991 (t(5,5)).

Sequence in context: A364406 A005597 A281056 * A197013 A329092 A081825

Adjacent sequences: A273986 A273987 A273988 * A273990 A273991 A273992

Keyword

cons,nonn

Author

Jean-François Alcover, Jun 06 2016

Status

approved