Glaisher-Kinkelin Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Define

$$K(n+1) \equiv 0^0 1^1 2^2 3^3\cdots n^n$$ (1)

$$G(n+1) \equiv {(n!)^n\over K(n+1)}=\left\{\begin{array}{ll} 1 & \mbox{if $n=0$}\\ 0! 1! 2! \cdots (n-1)! & \mbox{if $n>0$.}\end{array}\right.$$ (2)

where $G$ is the G-Function and $K$ is the K-Function. Then

$$\lim_{n\to\infty} {K(n+1)\over n^{n^2/2+n/2+1/2} e^{-n^2/4}}=A$$ (3)

$$\lim_{n\to\infty} {G(n+1)\over n^{n^2/2-1/12}(2\pi)^{n/2} e^{-3n^2/4}}={e^{1/12}\over A},$$ (4)

where

$$A=\mathop{\rm exp}\nolimits \left[{-{\zeta'(2)\over 2\pi^2}+{\ln(2\pi)\over 12}+{\gamma\over 2}}\right]= 1.28242713\ldots,$$ (5)

where $\zeta(z)$ is the Riemann Zeta Function, $\pi$ is Pi, and $\gamma$ is the Euler-Mascheroni Constant (Kinkelin 1860, Glaisher 1877, 1878, 1893, 1894). Glaisher (1877) also obtained

$$A=2^{7/36}\pi^{-1/6}\mathop{\rm exp}\nolimits \left\{{{1\over 3}+{2\over 3}\int_0^{1/2} \ln[\Gamma(x+1)]\,dx}\right\}.$$ (6)

Glaisher (1894) showed that

$$1^{1/1} 2^{1/4} 3^{1/9} 4^{1/16} 5^{1/25} \cdots =\left({A^{12}\over 2\pi e^\gamma}\right)^{\pi^2/6}$$ (7)

$$1^{1/1} 3^{1/9} 5^{1/25} 7^{1/49} 9^{1/81} \cdots = \left({A^{12}\over 2^{4/3} \pi e^\gamma}\right)^{\pi^2/8}$$ (8)

$${1^{1/1} 5^{1/125} 9^{1/729}\cdots\over 3^{1/27} 7^{1/343} 1... ...over 2^{5/32}\pi^{1/32}e^{3/32+\gamma/48+s/4}}\right)^{\pi^3},$$ (9)

where

$$s\equiv {\zeta(3)\over 3\cdot 4\cdot 5} {1\over 4^3}+{\zeta(... ...er 4^5} + {\zeta(7)\over 7\cdot 8\cdot 9}{1\over 4^7}+\ldots.$$ (10)

See also G-Function, Hyperfactorial, K-Function

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/glshkn/glshkn.html

Glaisher, J. W. L. ``On a Numerical Continued Product.'' Messenger Math. 6, 71-76, 1877.

Glaisher, J. W. L. ``On the Product $1^12^23^3\cdots n^n$.'' Messenger Math. 7, 43-47, 1878.

Glaisher, J. W. L. ``On Certain Numerical Products.'' Messenger Math. 23, 145-175, 1893.

Glaisher, J. W. L. ``On the Constant which Occurs in the Formula for $1^1 2^2 3^3\cdots\d{}n^n$.'' Messenger Math. 24, 1-16, 1894.

Kinkelin. ``Über eine mit der Gammafunktion verwandte Transcendente und deren Anwendung auf die Integralrechnung.'' J. Reine Angew. Math. 57, 122-158, 1860.