K-Function

An extension of the $K$-function

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

defined by

$$K(z)={[\Gamma(z)]^z\over G(z)}.$$ (2)

Here, $G(z)$ is the G-Function defined by

$$G(n+1)\equiv {(n!)^n\over K(n+1)}=\cases{ 1 & if $n=0$\cr 0! 1! 2! \cdots (n-1)! & if $n>0$.\cr}$$ (3)

The $K$-function is given by the integral

$$K(z)=(2\pi)^{-(z-1)/2}\mathop{\rm exp}\nolimits \left[{{z\choose 2}+\int_0^{z-1} \ln(t!)\,dt}\right]$$ (4)

and the closed-form expression

$$K(z)=\mathop{\rm exp}\nolimits [\zeta'(-1,z)-\zeta'(-1)],$$ (5)

where $\zeta(z)$ is the Riemann Zeta Function, $\zeta'(z)$ its Derivative, $\zeta(a,z)$ is the Hurwitz Zeta Function, and

$$\zeta'(a,z)\equiv \left[{d\zeta(s,z)\over ds}\right]_{s=a}.$$ (6)

$K(z)$ also has a Stirling-like series

$$K(z+1)=(2^{1/3}\pi_1 z)^{1/12} z^{{z+1\choose 2}}\mathop{\rm... ...dot 3\cdot 4z^2}-{B_6\over 4\cdot 5\cdot 6z^4}-\ldots}\right),$$ (7)

where

$$\pi_1 \equiv [K({\textstyle{1\over 2}})]^8$$ (8)

$$= e^{-(\ln 2)/3-12\zeta'(-1)}$$ (9)

$$= 2^{2/3}\pi e^{\gamma-1-\zeta'(2)/\zeta(2)},$$ (10)

and $\gamma$ is the Euler-Mascheroni Constant (Gosper).

The first few values of $K(n)$ for $n=2$, 3, ... are 1, 4, 108, 27648, 86400000, 4031078400000, ... (Sloane's A002109). These numbers are called Hyperfactorials by Sloane and Plouffe (1995).

See also G-Function, Glaisher-Kinkelin Constant, Hyperfactorial, Stirling's Series

References

Sloane, N. J. A. Sequence A002109/M3706 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.