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

0,1

Comments

Also known as the third Bendersky constant.

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights, 2011, p. 53, eq. (70).

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.

Junesang Choi and H. M. Srivastava, Certain classes of series associated with the Zeta function and multiple gamma functions, Journal of Mathematical Analysis and Applications, Vol. 118, No. 1-2 (2000), pp. 87-109.

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.

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

Equals exp(lim_{n->oo} Sum_{k=1..n} k^3*log(k) - (n^4/4 + n^3/2 + n^2/4 - 1/120)*log(n) + n^4/16 - n^2/12) (Choi and Srivastava, 2000). - Amiram Eldar, May 22 2026

Example

0.97955552694284460582421883726349...

Mathematica

RealDigits[Exp[-11/720 - Zeta'[-3]], 10, 98] // First
(* Alternative: *)
RealDigits[Exp[(BernoulliB[4]/4) * (EulerGamma + Log[2 * Pi] - (Zeta'[4]/Zeta[4]))], 10, 100] // First (* G. C. Greubel, Dec 31 2015 *)

Prog

(PARI) exp(-11/720 - zeta'(-3)) \\ Stefano Spezia, Dec 01 2024

Crossrefs

Cf. A001620, A255321, A259068.

Cf. A019727, A074962, A243262, A243263, A243264, A243265, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.

Sequence in context: A199003 A199048 A094132 * A244691 A121911 A358030

Adjacent sequences: A243260 A243261 A243262 * A243264 A243265 A243266

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 02 2014

Status

approved