cp's OEIS Frontend

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).

%I A273959 #20 Aug 02 2018 16:53:04
%S A273959 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 A273959 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 A273959 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 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).
%H A273959 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 A273959 C = Pi/16 (1 - 1/sqrt(5)) (1+2 Sum_{n>=1} exp(-n^2 Pi sqrt(15)))^4.
%F A273959 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 A273959 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 A273959 0.10854386983368497104035275675922632616425672443479475045864659238...
%t A273959 c = (Pi/16) (1 - 1/Sqrt[5]) EllipticTheta[3, 0, Exp[-Sqrt[15] Pi]]^4;
%t A273959 RealDigits[c, 10, 100][[1]]
%t A273959 RealDigits[((5 - Sqrt[5]) EllipticK[(16 - 7 Sqrt[3] - Sqrt[15])/32]^2)/(20 Pi), 10, 100][[1]] (* _Jan Mangaldan_, Jan 04 2017 *)
%t A273959 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 A273959 (PARI) th(x)=suminf(y=1, x^y^2)
%o A273959 (1-1/sqrt(5))*(1+2*th(exp(-sqrt(15)*Pi)))^4*Pi/16 \\ _Charles R Greathouse IV_, Jun 06 2016
%o A273959 (PARI) K(x)=Pi/2/agm(1,sqrt(1-x))
%o A273959 ((5 - sqrt(5))*K((16 - 7*sqrt(3) - sqrt(15))/32)^2)/20/Pi \\ _Charles R Greathouse IV_, Aug 02 2018
%K A273959 nonn,cons
%O A273959 0,3
%A A273959 _Jean-François Alcover_, Jun 05 2016