|
%I #31 Dec 15 2020 03:30:01
%S 8,0,9,3,9,6,5,9,7,3,6,6,2,9,0,1,0,9,5,7,8,6,8,0,4,7,8,7,2,6,3,8,2,1,
%T 1,9,3,7,2,7,8,7,6,4,8,2,6,1,1,3,0,1,6,5,8,7,7,5,8,3,3,2,4,9,0,8,8,1,
%U 4,9,1,1,3,7,3,6,2,2,7,8,9,3,7,4,6,0,1,8,3,3,8,5,7,3,5,3,0,1,4,6,2,7,1,2,6
%N Decimal expansion of Product_{n >= 2} 1-n^(-3).
%C The physical applications of this kind of product (with s<0) can be found in the Klauder et al. reference. - _Karol A. Penson_, Feb 24 2006
%H J. R. Klauder, K. A. Penson and J.-M. Sixdeniers, <a href="http://dx.doi.org/10.1103/PhysRevA.64.013817">Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems</a>, Physical Review A, Vol. 64, p. 013817 (2001).
%F Equals cosh((sqrt(3)*Pi)/2)/(3*Pi).
%F Product_{n >= 2} (1 - 1/n^p) simplifies, if p is odd, to 1/(p * Product(Gamma(- (-1)^(j*(1 + 1/p))), (j, 1, p - 1})) and, if p is even, to the elementary Product(Sin(Pi*(-1)^((2*j)/p))/(Pi*I), (j, 1, p/2 - 1}) / p. - David W. Cantrell, Feb 24 2006
%F Equals exp(Sum_{j>=1} (1 - zeta(3*j))/j). - _Vaclav Kotesovec_, Apr 27 2020
%F Equals 1/(Gamma((5-i*sqrt(3))/2)*Gamma((5+i*sqrt(3))/2)). - _Amiram Eldar_, Sep 01 2020
%e 0.809396597366290109578680478726382119372787648261130...
%t RealDigits[N[Cosh[(Sqrt[3]*Pi)/2]/(3*Pi), 150]] (* _T. D. Noe_, Apr 24 2006 *)
%o (PARI) exp(suminf(j=1, (1 - zeta(3*j))/j)) \\ _Vaclav Kotesovec_, Apr 27 2020
%Y Cf. A073017, A175615, A175617, A175619, A258870.
%K nonn,cons
%O 0,1
%A _Zak Seidov_, Apr 17 2006
%E Corrected and extended by _T. D. Noe_, Apr 24 2006
|
