K-function

In mathematics, the K-function, typically denoted "K"("z"), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the Gamma function.

Formally, the K-function is defined as

:

It can also be given in closed form as

:$K(z)=expleft\; [zeta^prime(-1,z)-zeta^prime(-1)\; ight]$

where ζ'("z") denotes the derivative of the Riemann zeta function, ζ("a","z") denotes the Hurwitz zeta function and

:$zeta^prime(a,z)\; stackrel\{mathrm\{def\{=\}\; left\; [frac\{dzeta(s,z)\}\{ds\}\; ight]\; \_\{s=a\}.$

The K-function is closely related to the Gamma function and the Barnes G-function; for natural numbers "n", we have

:$K(n)=frac\{(Gamma(n))^\{n-1\{G(n)\}.$

More prosaically, one may write

:$K(n+1)=1^1,\; 2^2,\; 3^3\; cdots\; n^n.$

References

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