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.

Definition
Formally, the $K$-function is defined as


 * $$K(z)=(2\pi)^{-\frac{z-1}2} \exp\left[\binom{z}{2}+\int_0^{z-1} \ln \Gamma(t + 1)\,dt\right].$$

It can also be given in closed form as


 * $$K(z)=\exp\bigl[\zeta'(-1,z)-\zeta'(-1)\bigr]$$

where $&zeta;′(z)$ denotes the derivative of the Riemann zeta function, $&zeta;(a,z)$ denotes the Hurwitz zeta function and


 * $$\zeta'(a,z)\ \stackrel{\mathrm{def}}{=}\ \left.\frac{\partial\zeta(s,z)}{\partial s}\right|_{s=a}.$$

Another expression using the polygamma function is


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

Or using the balanced generalization of the polygamma function:


 * $$K(z)=A \exp\left[\psi(-2,z)+\frac{z^2-z}{2}\right]$$

where $K$ is the Glaisher constant.

Similar to the Bohr-Mollerup Theorem for the gamma function, the log K-function is the unique (up to an additive constant) eventually 2-convex solution to the equation $$\Delta f(x)=x\ln(x)$$ where $$\Delta$$ is the forward difference operator.

Properties
It can be shown that for $α > 0$:


 * $$\int_\alpha^{\alpha+1}\ln K(x)\,dx-\int_0^1\ln K(x)\,dx=\tfrac{1}{2}\alpha^2\left(\ln\alpha-\tfrac{1}{2}\right)$$

This can be shown by defining a function $A$ such that:


 * $$f(\alpha)=\int_\alpha^{\alpha+1}\ln K(x)\,dx$$

Differentiating this identity now with respect to $f$ yields:


 * $$f'(\alpha)=\ln K(\alpha+1)-\ln K(\alpha)$$

Applying the logarithm rule we get


 * $$f'(\alpha)=\ln\frac{K(\alpha+1)}{K(\alpha)}$$

By the definition of the $α$-function we write


 * $$f'(\alpha)=\alpha\ln\alpha$$

And so


 * $$f(\alpha)=\tfrac12\alpha^2\left(\ln\alpha-\tfrac12\right)+C$$

Setting $α = 0$ we have


 * $$\int_0^1 \ln K(x)\,dx=\lim_{t\rightarrow0}\left[\tfrac12 t^2\left(\ln t-\tfrac12\right)\right]+C \ =C$$

Now one can deduce the identity above.

The $K$-function is closely related to the gamma function and the Barnes $K$-function; for natural numbers $G$, we have


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

More prosaically, one may write


 * $$K(n+1)=1^1 \cdot 2^2 \cdot 3^3 \cdots n^n.$$

The first values are
 * 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ....