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

There are multiple equivalent definitions of the K-function.

The direct definition:

$K(z)=(2\pi {)}^{-\frac{z-1}{2}}\exp [(\genfrac{}{}{0ex}{}{z}{2})+{\int }_0^{z-1}\ln \Gamma (t+1)dt].$

Definition via

$K(z)=\exp [{\zeta }^{\prime }(-1,z)-{\zeta }^{\prime }(-1)]$

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

${\zeta }^{\prime }(a,z)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\frac{\partial \zeta (s,z)}{\partial s}|}_{s=a},\zeta (s,q)=\sum _{k=0}^{\mathrm{\infty }}(k+q{)}^{-s}$

Definition via polygamma function:

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

Definition via balanced generalization of the polygamma function:

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

where A is the Glaisher constant.

It can be defined via unique characterization, similar to how the gamma function can be uniquely characterized by the Bohr-Mollerup Theorem:

Let $f:(0,\mathrm{\infty })\to ℝ$ be a solution to the functional equation $f(x+1)-f(x)=x\ln x$, such that there exists some $M>0$, such that given any distinct $x_0,x_1,x_2,x_3\in (M,\mathrm{\infty })$, the divided difference $f[x_0,x_1,x_2,x_3]\ge 0$. Such functions are precisely $f=\ln K+C$, where $C$ is an arbitrary constant.

Properties

For α > 0:

${\int }_{\alpha }^{\alpha +1}\ln K(x)dx-{\int }_0^1\ln K(x)dx=\frac{1}{2}{\alpha }^2(\ln \alpha -\frac{1}{2})$

Proof

Proof

Let $f(\alpha )={\int }_{\alpha }^{\alpha +1}\ln K(x)dx$

Differentiating this identity now with respect to α yields:

$f^{\prime }(\alpha )=\ln K(\alpha +1)-\ln K(\alpha )$

Applying the logarithm rule we get

$f^{\prime }(\alpha )=\ln \frac{K(\alpha +1)}{K(\alpha )}$

By the definition of the K-function we write

$f^{\prime }(\alpha )=\alpha \ln \alpha$

And so

$f(\alpha )=\frac{1}{2}{\alpha }^2(\ln \alpha -\frac{1}{2})+C$

Setting α = 0 we have

${\int }_0^1\ln K(x)dx=\underset{t\to 0}{lim}\left[\frac{1}{2}{t}^2(\ln t-\frac{1}{2})\right]+C=C$

Functional equations

The K-function is closely related to the gamma function and the Barnes G-function. For all complex $z$, $$K(z)G(z)=e^{(z-1)\ln \Gamma (z)}$$

Multiplication formula

Similar to the multiplication formula for the gamma function:

$\prod _{j=1}^{n-1}\Gamma \left(\frac{j}{n}\right)=\sqrt{\frac{(2\pi {)}^{n-1}}{n}}$

there exists a multiplication formula for the K-Function involving Glaisher's constant:

$\prod _{j=1}^{n-1}K\left(\frac{j}{n}\right)=A^{\frac{n^2-1}{n}}{n}^{-\frac{1}{12n}}{e}^{\frac{1-n^2}{12n}}$

Integer values

For all non-negative integers,$$K(n+1)=1^1\cdot 2^2\cdot 3^3\cdots n^n=H(n)$$where $H$ is the hyperfactorial.

The first values are

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