What is "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

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

It can also be given in closed form as

K(z) = exp[ζ( − 1, z) − ζ( − 1)]

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

$$\zeta^\prime(a,z)\ \stackrel{\mathrm{def}}{=}\ \left[\frac{\partial\zeta(s,z)}{\partial s}\right]_{s=a}.$$

Another expression using polygamma function is

$$K(z)=\exp\left(\psi^{(-2)}(z)+\frac{z^2-z}{2}-\frac z2 \ln (2\pi)\right)$$

Or using balanced generalization of Polygamma function:

$$K(z)=A e^{\psi(-2,z)+\frac{z^2-z}{2}}$$

where A is Glaisher constant.

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 2 3⋯n.

The first values are

1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... .