K-function
From Wikipedia, the free encyclopedia
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].](../../../math/9/9/1/9912baf438ae03d70d8a286bd016c082.png)
It can also be given in closed form as
![K(z)=\exp\left[\zeta^\prime(-1,z)-\zeta^\prime(-1)\right]](../../../math/8/a/5/8a540da338ac595b899a0d2015882ac3.png)
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}.](../../../math/2/b/4/2b4d4c6cadc0a5e6a8e5c0ca6f2b0131.png)
The K-function is closely related to the Gamma function and the Barnes G-function; for natural numbers n, we have
More prosaically, one may write
[edit] References
- Eric W. Weisstein, K-Function at MathWorld.
