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

Author

Jean-François Alcover, Jun 05 2016

Keywords

• nonn
• cons

Examples

			0.10854386983368497104035275675922632616425672443479475045864659238...
		

Links

• 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.

Programs

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 *)

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

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.