A273959 - OEIS

%I #20 Aug 02 2018 16:53:04

%S 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,

%T 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,

%U 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

%N 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).

%H David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, <a href="http://arxiv.org/abs/0801.0891">Elliptic integral evaluations of Bessel moments</a>, arXiv:0801.0891 [hep-th], 2008, page 21.

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

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

%F 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.

%e 0.10854386983368497104035275675922632616425672443479475045864659238...

%t c = (Pi/16) (1 - 1/Sqrt[5]) EllipticTheta[3, 0, Exp[-Sqrt[15] Pi]]^4;

%t RealDigits[c, 10, 100][[1]]

%t RealDigits[((5 - Sqrt[5]) EllipticK[(16 - 7 Sqrt[3] - Sqrt[15])/32]^2)/(20 Pi), 10, 100][[1]] (* _Jan Mangaldan_, Jan 04 2017 *)

%t 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 *)

%o (PARI) th(x)=suminf(y=1, x^y^2)

%o (1-1/sqrt(5))*(1+2*th(exp(-sqrt(15)*Pi)))^4*Pi/16 \\ _Charles R Greathouse IV_, Jun 06 2016

%o (PARI) K(x)=Pi/2/agm(1,sqrt(1-x))

%o ((5 - sqrt(5))*K((16 - 7*sqrt(3) - sqrt(15))/32)^2)/20/Pi \\ _Charles R Greathouse IV_, Aug 02 2018

%K nonn,cons

%O 0,3

%A _Jean-François Alcover_, Jun 05 2016