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
Examples
1.2824271291006226368753425688697917277676889273250011920637400217404...
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
- L. Almodovar, V. H. Moll, H. Quand, Infinite products arising in paperfolding, JIS 19 (2016) # 16.5.1 eq. (13)
- 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.
Crossrefs
Programs
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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
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Mathematica
RealDigits[Glaisher, 10, 111][[1]] (* Robert G. Wilson v *)
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PARI
x=10^(-100); exp(1/12-(zeta(-1+x)-zeta(-1))/x)
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PARI
exp(1/12-zeta'(-1)) \\ Charles R Greathouse IV, Dec 12 2013
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)
Extensions
More terms from Sascha Kurz, Feb 03 2003
Comments