A109219 Decimal expansion of Product_{n >= 2} 1-n^(-3).
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, 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, 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
Offset: 0
Examples
0.809396597366290109578680478726382119372787648261130...
Links
- J. R. Klauder, K. A. Penson and J.-M. Sixdeniers, Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems, Physical Review A, Vol. 64, p. 013817 (2001).
Programs
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Mathematica
RealDigits[N[Cosh[(Sqrt[3]*Pi)/2]/(3*Pi), 150]] (* T. D. Noe, Apr 24 2006 *)
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PARI
exp(suminf(j=1, (1 - zeta(3*j))/j)) \\ Vaclav Kotesovec, Apr 27 2020
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PARI
prodnumrat(1-1/x^3,2) \\ Charles R Greathouse IV, Feb 03 2025
Formula
Equals cosh((sqrt(3)*Pi)/2)/(3*Pi).
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
Equals exp(Sum_{j>=1} (1 - zeta(3*j))/j). - Vaclav Kotesovec, Apr 27 2020
Equals 1/(Gamma((5-i*sqrt(3))/2)*Gamma((5+i*sqrt(3))/2)). - Amiram Eldar, Sep 01 2020
Extensions
Corrected and extended by T. D. Noe, Apr 24 2006
Comments