A074962 Decimal expansion of Glaisher-Kinkelin constant A.

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

Offset

1,2

Comments

Arises in expressions such as A002109(n) = 1^1*2^2*3^3*...*n^n which is asymptotic to A*n^(n^2/2 + n/2 + 1/12)*exp(-n^2/4). See A002109 for more references and links.

Named after the English mathematician and astronomer James Whitbread Lee Glaisher (1848-1928) and the Swiss mathematician Hermann Kinkelin (1832-1913). - Amiram Eldar, Jun 15 2021

References

Steven R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, p. 135.

Konrad Knopp, Theory and applications of infinite series, Dover, p. 555.

Links

Gheorghe Coserea, Table of n, a(n) for n = 1..10010

Leyda Almodovar, Victor H. Moll, Hadrian Quand, Fernando Roman, Eric Rowland, and Michole Washington, Infinite products arising in paperfolding, JIS 19 (2016), Article 16.5.1, eq. (13).

Ernest William Barnes, The theory of the G-function, Q. J. Pure Appl. Math., Vol. 31 (1900), pp. 264-314. See pp. 266-267.

Chao-Ping Chen and Long Lin, Asymptotic expansions related to Glaisher-Kinkelin constant based on the Bell polynomials, Journal of Number Theory, Vol. 133 (2013), pp. 2699-2705.

Ovidiu Furdui, proposer, Problem 11494, Amer. Math. Monthly, Vol. 118, No. 9 (2011), 850-852.

J. W. L. Glaisher, On the Product 1^1.2^2.3^3...n^n, The Messenger of Mathematics, Vol. 7 (1878), pp. 43-47.

Antonio Gracia Llorente, A Simple Limit-Product Formula for Glaisher’s Constant, OSF Preprint, 2025.

Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J., Vol. 16 (2008), pp. 247-270; see Examples 5.2, 5.7, 5.11.

Fredrik Johansson et al., mpmath, Mathematical constants (Mpmath).

Fredrik Johansson et al., mpmath, Glaisher's constant to 20,000 digits.

Hermann Kinkelin, Über eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung, Journal für die reine und angewandte Mathematik, Vol. 57 (1860), pp. 122-138.

Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., Vol. 332, No. 1 (2007), pp. 292-314; see Section 5.

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 World of Mathematics, Glaisher-Kinkelin Constant.

Wikipedia, Glaisher-Kinkelin constant.

Formula

A = 2^(1/36)*Pi^(1/6)*exp(1/3*(-Gamma/4 + s(2)/3 - s(3)/4 + ...)) where s(k) denotes Sum_{n>=0} 1/(2n+1)^k.

Closed expressions for A are exp(-zeta'(2)/2/Pi^2 + log(2*Pi)/12 + Gamma/12) or exp(1/12-zeta'(-1)).

Equals (2*Pi)^(1/4) / limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2). - Vaclav Kotesovec, Dec 02 2023

Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^4-1)))^c, where gamma is Euler's constant (A001620), and c = Bernoulli(2)/2 = 1/12 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024

Equals e^(-1/4 + Integral_{x=1..2} x*log(sqrt(2*Pi)) - B_2(x) + x^2*Psi(x)/2 dx), where B_2(x) is the second Bernoulli polynomial and Psi(x) is the digamma function. - Andrea Pinos, Apr 16 2024

Equals exp(1/12 - 2*Integral_{x=0..oo} x*log(x)/(exp(2*Pi*x) - 1) dx) = exp(1/3 + 7*log(2)/36 - log(Pi)/6 + (2/3)*Integral_{x=0..1/2} log(Gamma(x+1)) dx) (see Finch). - Stefano Spezia, Dec 01 2024

From Antonio Graciá Llorente, May 03 2025: (Start)

Equals lim_{n->oo} (2^(13/3)*n)^(1/12) * Product_{k=1..n} (1 - 1/(2*k+1)^2)^((2*k+1)/6).

Equals lim_{n->oo} (24*n^2)^(1/24) * Product_{prime p<=n} (p^(1 - p/(p^2-1)) / sqrt(p^2-1))^(1/12). (End)

Equals 2^(1/36) * exp(1/12)/(Pi^(1/6) * G(1/2)^(2/3)), where G is the Barnes G-function (Barnes, 1900, p. 294). - Amiram Eldar, May 22 2026

From Vaclav Kotesovec, May 22 2026: (Start)

Equals exp(1/12 - 2*G/(9*Pi)) / (BarnesG(1/4)^(8/9) * Gamma(1/4)^(2/3)).

Equals exp(1/12 + 2*G/(9*Pi)) / (BarnesG(3/4)^(8/9) * Gamma(3/4)^(2/9)), where G is Catalan's constant (A006752). (End)

Example

1.2824271291006226368753425688697917277676889273250011920637400217404...

Maple

evalf(limit(product(k^k, k=1..n)/(n^(n^2/2+n/2+1/12)*exp(-n^2/4)), n=infinity), 120); # Vaclav Kotesovec, Oct 23 2014

Mathematica

RealDigits[Glaisher, 10, 111][[1]] (* Robert G. Wilson v, Jan 26 2011 *)

Prog

(PARI) my(x=10^(-100)); exp(1/12-(zeta(-1+x)-zeta(-1))/x)
(PARI) exp(1/12-zeta'(-1)) \\ Charles R Greathouse IV, Dec 12 2013

Crossrefs

Cf. A001620, A006752, A243262, A243263, A243264, A243265.

Cf. A000178, A002109, A051675, A255321, A255323, A255344.

Sequence in context: A296049 A388166 A388825 * A064863 A021358 A332353

Adjacent sequences: A074959 A074960 A074961 * A074963 A074964 A074965

Keyword

nonn,cons,nice

Author

Benoit Cloitre, Oct 05 2002

Extensions

More terms from Sascha Kurz, Feb 03 2003

Status

approved