A243263 Decimal expansion of the generalized Glaisher-Kinkelin constant A(3).
9, 7, 9, 5, 5, 5, 5, 2, 6, 9, 4, 2, 8, 4, 4, 6, 0, 5, 8, 2, 4, 2, 1, 8, 8, 3, 7, 2, 6, 3, 4, 9, 1, 8, 2, 6, 4, 4, 5, 5, 3, 6, 7, 5, 2, 4, 9, 5, 5, 2, 9, 9, 0, 2, 2, 5, 7, 7, 1, 7, 1, 4, 2, 7, 9, 7, 5, 8, 8, 5, 6, 7, 2, 4, 8, 1, 5, 5, 9, 6, 1, 4, 9, 4, 4, 4, 4, 4, 3, 5, 3, 8, 3, 3, 2, 1, 9, 6
Offset: 0
Examples
0.97955552694284460582421883726349...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
- Victor S. Adamchik, Polygamma functions of negative order, Journal of Computational and Applied Mathematics, Vol. 100, No. 2 (1998), pp. 191-199.
- L. Bendersky, Sur la fonction gamma généralisée, Acta Mathematica , Vol. 61 (1933), pp. 263-322; alternative link.
- Robert A. Van Gorder, Glaisher-type products over the primes, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
- Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant.
Crossrefs
Programs
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Mathematica
RealDigits[Exp[-11/720 - Zeta'[-3]], 10, 98] // First RealDigits[Exp[(BernoulliB[4]/4) * (EulerGamma + Log[2 * Pi] - (Zeta'[4]/Zeta[4]))], 10, 100] // First (* G. C. Greubel, Dec 31 2015 *)
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PARI
exp(-11/720 - zeta'(-3)) \\ Stefano Spezia, Dec 01 2024
Formula
A(k) = exp(B(k+1)/(k+1)*H(k) - zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(3) = exp(-11/720 - zeta'(-3)).
Equals exp(3*zeta'(4)/(4*Pi^4) - gamma/120) / (2*Pi)^(1/120), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 24 2015
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^4-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(4)/4 = -1/120 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
Comments