K-function

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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

K(z)=(2\pi)^{(-z-1)/2} \exp\left[\begin{pmatrix} z\\ 2\end{pmatrix}+\int_0^{z-1} \ln(t!)\,dt\right].

It can also be given in closed form as

K(z)=\exp\left[\zeta^\prime(-1,z)-\zeta^\prime(-1)\right]

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{d\zeta(s,z)}{ds}\right]_{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.

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