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URL: https://oeis.org/A273959

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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).
6
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
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: A030437 A200290 A382259 * A010525 A190184 A275984
KEYWORD
nonn,cons
AUTHOR
STATUS
approved