|
%I #95 Feb 08 2024 01:57:36
%S 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,
%T 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,
%U 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
%N Decimal expansion of Glaisher-Kinkelin constant A.
%C 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.
%C 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
%D Steven R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, p. 135.
%D Konrad Knopp, Theory and applications of infinite series, Dover, p. 555.
%H Gheorghe Coserea, <a href="/A074962/b074962.txt">Table of n, a(n) for n = 1..10010</a>
%H Chao-Ping Chen and Long Lin, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.011">Asymptotic expansions related to Glaisher-Kinkelin constant based on the Bell polynomials</a>, Journal of Number Theory, Vol. 133 (2013), pp. 2699-2705.
%H Ovidiu Furdui, proposer, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.09.846">Problem 11494</a>, Amer. Math. Monthly, Vol. 118, No. 9 (2011), 850-852.
%H J. W. L. Glaisher, <a href="http://www.archive.org/stream/messengermathem01glaigoog#page/n57/mode/1up">On the Product 1^1.2^2.3^3...n^n</a>, The Messenger of Mathematics, Vol. 7 (1878), pp. 43-47.
%H Jesús Guillera and Jonathan Sondow, <a href="http://arXiv.org/abs/math.NT/0506319">Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent</a>, Ramanujan J., Vol. 16 (2008), pp. 247-270; see Examples 5.2, 5.7, 5.11.
%H Fredrik Johansson et al., mpmath, <a href="http://mpmath.org/doc/current/functions/constants.html">Mathematical constants (Mpmath)</a>.
%H Fredrik Johansson et al., mpmath, <a href="https://www.webcitation.org/6BoWvFMX1?url=http://mpmath.googlecode.com/svn/data/glaisher.txt">Glaisher's constant to 20,000 digits</a>.
%H Hermann Kinkelin, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002150824&IDDOC=266726">Über eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung</a>, Journal für die reine und angewandte Mathematik, Vol. 57 (1860), pp. 122-138.
%H Jonathan Sondow and Petros Hadjicostas, <a href="http://dx.doi.org/10.1016/j.jmaa.2006.09.081">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl., Vol. 332, No. 1 (2007), pp. 292-314; see Section 5.
%H Robert A. Van Gorder, <a href="https://doi.org/10.1142/S1793042112500297">Glaisher-type products over the primes</a>, International Journal of Number Theory, Vol. 8, No. 2 (2012), pp. 543-550.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin Constant</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Glaisher%E2%80%93Kinkelin_constant">Glaisher-Kinkelin constant</a>.
%F 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.
%F Closed expressions for A are exp(-zeta'(2)/2/Pi^2 + log(2*Pi)/12 + Gamma/12) or exp(1/12-zeta'(-1)).
%F Equals (2*Pi)^(1/4) / limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2). - _Vaclav Kotesovec_, Dec 02 2023
%F 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
%e 1.2824271291006226368753425688697917277676889273250011920637400217404...
%p 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
%t RealDigits[Glaisher, 10, 111][[1]] (* _Robert G. Wilson v_ *)
%o (PARI) x=10^(-100); exp(1/12-(zeta(-1+x)-zeta(-1))/x)
%o (PARI) exp(1/12-zeta'(-1)) \\ _Charles R Greathouse IV_, Dec 12 2013
%Y Cf. A001620, A243262, A243263, A243264, A243265.
%Y Cf. A000178, A002109, A051675, A255321, A255323, A255344.
%K nonn,cons,nice
%O 1,2
%A _Benoit Cloitre_, Oct 05 2002
%E More terms from _Sascha Kurz_, Feb 03 2003
|
