A273990 -id:A273990 - cp's OEIS frontend

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

Jean-François Alcover, Jun 06 2016

			0.660344869018672357837266831705994263854241991696873858300803587553894...

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.

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)).

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 *)

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))).

A273991 Decimal expansion of the odd Bessel moment t(5,5) (see the referenced paper about the elliptic integral evaluations of Bessel moments).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 06 2016

			1.0432973673868713449189316078947712217566163312269155788688325589866...

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.

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

t[5, 5] = NIntegrate[x^5*BesselI[0, x]^2*BesselK[0, x]^3, {x, 0, Infinity}, WorkingPrecision -> 103];
RealDigits[t[5, 5]][[1]]

Integral_{0..inf} x^5*BesselI_0(x)^2*BesselK_0(x)^3.

Conjecture: Equals 76/15 t(5,3) - 16/45 t(5,1).

cp's OEIS Frontend

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

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