K-function

For the k-function, see Bateman 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

$K(z)=(2\pi)^{(-z%2B1)/2} \exp\left[\begin{pmatrix} z\\ 2\end{pmatrix}%2B\int_0^{z-1} \ln(t!)\,dt\right].$

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

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

Another expression using polygamma function is^[1]

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

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

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

$K(n%2B1)=1^1\, 2^2\, 3^3 \cdots n^n.$

The first values are

1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... ((sequence A002109 in OEIS)).

References