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Glaisher-Kinkelin Constant

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The Glaisher-Kinkelin constant A is defined by

lim_(n->infty)(H(n))/(n^(n^2/2+n/2+1/12)e^(-n^2/4))=A
(1)

(Glaisher 1878, 1894, Voros 1987), where H(n) is the hyperfactorial, as well as

lim_(n->infty)(G(n+1))/(n^(n^2/2-1/12)(2pi)^(n/2)e^(-3n^2/4))=(e^(1/12))/A,
(2)

where G(n) is the Barnes G-function.

It has closed-form representations

A=e^(1/(12)-zeta^'(-1))
(3)
=(2pi)^(1/12)[e^(gammapi^2/6-zeta^'(2))]^(1/(2pi^2))
(4)
=1.28242712...
(5)

(OEIS A074962), where gamma is the Euler-Mascheroni constant and zeta^'(z) is the derivative of the Riemann zeta function (Kinkelin 1860; Jeffrey 1862; Glaisher 1877, 1878, 1893, 1894; Voros 1987).

The constant A is implemented as Glaisher, and appears in a number of sums and integrals, especially those involving gamma functions and zeta functions.

Definite integrals include

int_0^(1/2)lnGamma(x+1)dx=-1/2-7/(24)ln2+1/4lnpi+3/2lnA
(6)
int_0^infty(xlnx)/(e^(2pix)-1)dx=1/(24)-1/2lnA
(7)

(Glaisher 1878; Almkvist 1998; Finch 2003, p. 135), where lnGamma(z) is the log gamma function.

Glaisher (1894) showed that

product_(k=1)^(infty)k^(1/k^2)=1^(1/1)2^(1/4)3^(1/9)4^(1/16)5^(1/25)...
(8)
=((A^(12))/(2pie^gamma))^(pi^2/6)
(9)
product_(k=1,3,5,...)^(infty)k^(1/k^2)=1^(1/1)3^(1/9)5^(1/25)7^(1/49)9^(1/81)...
(10)
=((A^(36))/(2^4pi^3e^(3gamma)))^(pi^2/24)
(11)

(OEIS A115521 and A115522; Glaisher 1894).

It also arises in the identity

sum_(k=2)^(infty)(lnk)/(k^2)=-zeta^'(2)
(12)
=1/6pi^2[12lnA-gamma-ln(2pi)]
(13)
=0.93754825431...
(14)
sum_(k=3,5,...)^(infty)(lnk)/(k^2)=pi^2(3/2lnA-1/6ln2-1/8lnpi-1/8gamma)
(15)

(OEIS A073002; Glaisher 1894), which follows from the above products.

Guillera and Sondow (2008) give

lnA=1/8+sum_(n=0)^infty1/(2(n+1))sum_(k=0)^n(-1)^(k+1)(n; k)(k+1)^2ln(k+1).
(16)

Another more spectacular product is

product_(k=1)^(infty)((4k+1)^(1/(4k+1)^3))/((4k-1)^(1/(4k-1)^3))=(A/(2^(5/32)pi^(1/32))e^(-3/32-gamma/48+p/4))^(pi^3)
(17)
=(2pi)^(-pi^3/32)e^({3pizeta(3)+pi^3[3-2gamma-128zeta^'(-2,1/4)]}/64)
(18)
=e^(-beta^'(3)),
(19)

where beta(z) is the Dirichlet beta function and

p=sum_(k=3,5,...)^(infty)(zeta(k))/(4^kk(k+1)(k+2))
(20)
=9/(16)-gamma/(24)+(ln2)/2-4lnA+(3zeta(3))/(16pi^2)-8zeta^'(-2,1/4)
(21)
=3/8+gamma/(12)-(4beta^'(3))/(pi^3)+(5ln2)/8-4lnA+(lnpi)/8
(22)

(Glaisher 1894).

It is also given by

A=2^(1/36)pi^(1/6)e^((-gamma/4+s)/3),
(23)

where

s=sum_(r=2)^(infty)((-1/2)^r(2^r-1)zeta(r))/(1+r)
(24)
=1/(12)[3+3gamma-36zeta^'(-1)-ln2-6lnpi]
(25)

(Glaisher 1878, 1894, who, however, did not obtain the closed form of this expression).

phi^(-1)<A<phi, where phi is the golden ratio, so A has a reciprocal proportion triangle.

See also

Barnes G-Function, Glaisher-Kinkelin Constant Continued Fraction, Glaisher-Kinkelin Constant Digits, Hyperfactorial, K-Function, MRB Constant

Related Wolfram sites

https://functions.wolfram.com/Constants/Glaisher/

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References

Almkvist, G. "Asymptotic Formulas and Generalized Dedekind Sums." Experim. Math. 7, 343-356, 1998.Finch, S. R. "Glaisher-Kinkelin Constant." §2.15 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 135-145, 2003.Glaisher, J. W. L. "On the Product 1^1.2^2.3^3...n^n." Messenger Math. 7, 43-47, 1878.Glaisher, J. W. L. "On Certain Numerical Products in Which the Exponents Depend upon the Numbers." Messenger Math. 23, 145-175, 1893.Glaisher, J. W. L. "On the Constant Which Occurs in the Formula for 1^1.2^2.3^3...n^n." Messenger Math. 24, 1-16, 1894.Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants via Analytic Continuations of Lerch's Transcendent." Ramanujan J. 16, 247-270, 2008. https://doi.org/10.1007/s11139-007-9102-0.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 88 and 113, 2003.Jeffrey, H. M. "On the Expansion of Powers of the Trigonometrical Ratios in Terms of Series of Ascending Powers of the Variables." Messenger Math. 5, 91-108, 1862.Kinkelin. "Über eine mit der Gammafunktion verwandte Transcendente und deren Anwendung auf die Integralrechnung." J. reine angew. Math. 57, 122-158, 1860.Sloane, N. J. A. Sequences A073002, A074962, A087501, A099791, A099792, A115521, and A115522 in "The On-Line Encyclopedia of Integer Sequences."Voros, A. "Spectral Functions, Special Functions and the Selberg Zeta Function." Commun. Math. Phys. 110, 439-465, 1987.

Referenced on Wolfram|Alpha

Glaisher-Kinkelin Constant

Cite this as:

Weisstein, Eric W. "Glaisher-Kinkelin Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Glaisher-KinkelinConstant.html

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