K-function

In mathematics, the Template:Mvar-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 Template:Mvar-function is defined as

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

It can also be given in closed form as

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

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

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

Another expression using the polygamma function is

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

Or using the balanced generalization of the polygamma function:

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

where Template:Mvar 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 $\mathrm{\Delta }f(x)=x\ln (x)$ where $\mathrm{\Delta }$ is the forward difference operator.

Properties

It can be shown that for α > 0:

$\underset{\alpha }{\overset{\alpha +1}{\int }}\ln K(x)dx-\underset{0}{\overset{1}{\int }}\ln K(x)dx=\frac{1}{2}{\alpha }^2(\ln \alpha -\frac{1}{2})$

This can be shown by defining a function Template:Mvar such that:

$f(\alpha )=\underset{\alpha }{\overset{\alpha +1}{\int }}\ln K(x)dx$

Differentiating this identity now with respect to Template:Mvar 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 Template:Mvar-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

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

Now one can deduce the identity above.

The Template:Mvar-function is closely related to the gamma function and the [[Barnes G-function|Barnes Template:Mvar-function]]; for natural numbers Template:Mvar, we have

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

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, ... (sequence A002109 in the OEIS).

Similar to the multiplication formula for the gamma function:

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

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

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