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\begin{figure}\begin{center}\BoxedEPSF{KFunctionReIm.epsf scaled 680}\end{center}\end{figure}

An extension of the $K$-function

K(n+1)\equiv 0^0 1^1 2^2 3^3\cdots n^n
\end{displaymath} (1)

defined by
K(z)={[\Gamma(z)]^z\over G(z)}.
\end{displaymath} (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}
\end{displaymath} (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]
\end{displaymath} (4)

and the closed-form expression
K(z)=\mathop{\rm exp}\nolimits [\zeta'(-1,z)-\zeta'(-1)],
\end{displaymath} (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}.
\end{displaymath} (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... 3\cdot 4z^2}-{B_6\over 4\cdot 5\cdot 6z^4}-\ldots}\right),
\end{displaymath} (7)

$\displaystyle \pi_1$ $\textstyle \equiv$ $\displaystyle [K({\textstyle{1\over 2}})]^8$ (8)
  $\textstyle =$ $\displaystyle e^{-(\ln 2)/3-12\zeta'(-1)}$ (9)
  $\textstyle =$ $\displaystyle 2^{2/3}\pi e^{\gamma-1-\zeta'(2)/\zeta(2)},$ (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


Sloane, N. J. A. Sequence A002109/M3706 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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© 1996-9 Eric W. Weisstein