A273959 Decimal expansion of 'C', an auxiliary constant defined by D. Broadhurst and related to Bessel moments (see the referenced paper about the elliptic integral evaluations of Bessel moments).

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

Offset

0,3

Links

Table of n, a(n) for n=0..99.

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

C = Pi/16 (1 - 1/sqrt(5)) (1+2 Sum_{n>=1} exp(-n^2 Pi sqrt(15)))^4.

Equals Pi/16 (1 - 1/sqrt(5)) theta_3(0, exp(-sqrt(15)*Pi))^4, where theta_3 is the elliptic theta_3 function.

Also equals s(5,1)/Pi^2 (where the Bessel moment s(5,1) is Integral_{0..inf} x I_0(x) K_0(x)^4 dx), a conjectural equality checked by the authors to 1200 decimal places.

Example

0.10854386983368497104035275675922632616425672443479475045864659238...

Mathematica

c = (Pi/16) (1 - 1/Sqrt[5]) EllipticTheta[3, 0, Exp[-Sqrt[15] Pi]]^4;
RealDigits[c, 10, 100][[1]]
RealDigits[((5 - Sqrt[5]) EllipticK[(16 - 7 Sqrt[3] - Sqrt[15])/32]^2)/(20 Pi), 10, 100][[1]] (* Jan Mangaldan, Jan 04 2017 *)
RealDigits[(Gamma[1/15] Gamma[2/15] Gamma[4/15] Gamma[8/15])/(240 Sqrt[5] Pi^2), 10, 100][[1]] (* Jan Mangaldan, Jan 04 2017 *)

Prog

(PARI) th(x)=suminf(y=1, x^y^2)
(1-1/sqrt(5))*(1+2*th(exp(-sqrt(15)*Pi)))^4*Pi/16 \\ Charles R Greathouse IV, Jun 06 2016
(PARI) K(x)=Pi/2/agm(1, sqrt(1-x))
((5 - sqrt(5))*K((16 - 7*sqrt(3) - sqrt(15))/32)^2)/20/Pi \\ Charles R Greathouse IV, Aug 02 2018

Crossrefs

Sequence in context: A134973 A030437 A200290 * A010525 A190184 A275984

Adjacent sequences: A273956 A273957 A273958 * A273960 A273961 A273962

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 05 2016

Status

approved