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Decimal expansion of Product_{n >= 2} 1-n^(-3).
12

%I #37 Feb 03 2025 12:25:32

%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_{j=1..p-1} Gamma(-(-1)^(j*(1 + 1/p)))) and, if p is even, to the elementary (Product_{j=1..p/2-1} sin(Pi*(-1)^(2*j/p))/(Pi*i)) / 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

%o (PARI) prodnumrat(1-1/x^3,2) \\ _Charles R Greathouse IV_, Feb 03 2025

%Y Cf. A073017, A175615, A175617, A175619, A258870.

%K nonn,cons,changed

%O 0,1

%A _Zak Seidov_, Apr 17 2006

%E Corrected and extended by _T. D. Noe_, Apr 24 2006