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A074962 Decimal expansion of Glaisher-Kinkelin constant A. 441
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 (list; constant; graph; refs; listen; history; text; internal format)
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
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.
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.
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 Product_{k>=1} 2^(10^(-k) + 3/13^k)((2*k)/(2*k + 1))^((k/3 + 1/12))((2*k + 2)/(2*k + 1))^((k/3 + 1/4)). - Antonio Graciá Llorente, May 20 2024
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 *)
PROG
(PARI) 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
Sequence in context: A257579 A199715 A296049 * A064863 A021358 A332353
KEYWORD
nonn,cons,nice
AUTHOR
Benoit Cloitre, Oct 05 2002
EXTENSIONS
More terms from Sascha Kurz, Feb 03 2003
STATUS
approved

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Last modified July 23 16:47 EDT 2024. Contains 374552 sequences. (Running on oeis4.)