A273951 - OEIS

A273951

Decimal expansion the even Bessel moment s(4,0) = Integral_{0..inf} I_0(x) K_0(x)^3 dx.

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

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OFFSET

1,1

LINKS

Table of n, a(n) for n=1..103.

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891, page 19.

R. J. Mathar, Some definite integrals over a power multiplied by four Modified Bessel Functions, vixra:1606.0141 (2016) eq. (35).

FORMULA

s(4,0) = Integral_{0..Pi/4) 4 EllipticK(-tan(x)^2) EllipticK(-cot(x)^2) / sin(2x) dx, where EllipticK is the complete elliptic integral of the first kind.

N.B. K(k) used in the paper is related to Mathematica's EllipticK(k) by K(k) = EllipticK(k^2/(k^2-1))/sqrt(1 - k^2).

EXAMPLE

6.997563016680632359556757826853096005697754284353362908336255807...

MATHEMATICA

s[4, 0] = NIntegrate[4 EllipticK[-Cot[t]^2] EllipticK[-Tan[t]^2] /Sin[2 t], {t, 0, Pi/4}, WorkingPrecision -> 103];

RealDigits[s[4, 0]][[1]]

CROSSREFS

Cf. A222068 (odd moment s(4,1)).

Sequence in context: A051496 A330564 A195296 * A307053 A100403 A066002

Adjacent sequences: A273948 A273949 A273950 * A273952 A273953 A273954

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Jun 05 2016

STATUS

approved