A243262 Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).

1, 0, 3, 0, 9, 1, 6, 7, 5, 2, 1, 9, 7, 3, 9, 2, 1, 1, 4, 1, 9, 3, 3, 1, 3, 0, 9, 6, 4, 6, 6, 9, 4, 2, 2, 9, 0, 6, 3, 3, 1, 9, 4, 3, 0, 6, 4, 0, 3, 4, 8, 7, 0, 6, 0, 2, 2, 7, 2, 6, 1, 7, 4, 1, 1, 4, 5, 1, 6, 6, 0, 6, 6, 9, 7, 8, 2, 9, 0, 4, 0, 5, 2, 9, 2, 9, 3, 1, 3, 6, 2, 5, 5, 4, 8, 0, 8, 8, 5

Offset: 1

Author

Jean-François Alcover, Jun 02 2014

Keywords

• nonn
• cons

Comments

Also known as the second Bendersky constant.

This is likely the same as the constant B considered in section 3 of the Choi and Srivastava link. - R. J. Mathar, Oct 03 2016

Examples

			1.03091675219739211419331309646694229...
		

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 = 1..2002
• J. Choi and H. M. Srivastava, Certain classes of series involving the zeta function, J. Math. Annal. Applic. 231 (1999) 91-117.
• K. Kimoto, N. Kurokawa, C. Sonoki, M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.
• Tobias Kyrion, A closed-form expression for zeta(3), arXiv:2008.05573 [math.GM], 2020.
• Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant

Crossrefs

Cf. A051675, A055462, A240966, A255269.

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

Cf. A002117 (zeta(3)).

Programs

Mathematica

RealDigits[Exp[Zeta[3]/(4*Pi^2)], 10, 99] // First
RealDigits[Exp[N[(BernoulliB[2]/4)*(Zeta[3]/Zeta[2]), 200]]]//First (* G. C. Greubel, Dec 31 2015 *)

PARI

exp(zeta(3)/(4*Pi^2)) \\ Felix Fröhlich, Jun 27 2019

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(0) = sqrt(2*Pi) (A019727),

A(1) = A = Glaisher-Kinkelin constant (A074962),

A(2) = exp(-zeta'(-2)) = exp(zeta(3)/(4*Pi^2)).

Equals exp(-A240966). - Vaclav Kotesovec, Feb 22 2015