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

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

It can also be given in closed form as

[math]\displaystyle{ K(z)=\exp\bigl[\zeta'(-1,z)-\zeta'(-1)\bigr] }[/math]

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

[math]\displaystyle{ \zeta'(a,z)\ \stackrel{\mathrm{def}}{=}\ \left.\frac{\partial\zeta(s,z)}{\partial s}\right|_{s=a}. }[/math]

Another expression using the polygamma function is

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

Or using the balanced generalization of the polygamma function:

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

where A 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 [math]\displaystyle{ \Delta f(x)=x\ln(x) }[/math] where [math]\displaystyle{ \Delta }[/math] is the forward difference operator.

Properties

It can be shown that for α > 0:

[math]\displaystyle{ \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) }[/math]

This can be shown by defining a function f such that:

[math]\displaystyle{ f(\alpha)=\int_\alpha^{\alpha+1}\ln K(x)\,dx }[/math]

Differentiating this identity now with respect to α yields:

[math]\displaystyle{ f'(\alpha)=\ln K(\alpha+1)-\ln K(\alpha) }[/math]

Applying the logarithm rule we get

[math]\displaystyle{ f'(\alpha)=\ln\frac{K(\alpha+1)}{K(\alpha)} }[/math]

By the definition of the K-function we write

[math]\displaystyle{ f'(\alpha)=\alpha\ln\alpha }[/math]

And so

[math]\displaystyle{ f(\alpha)=\tfrac12\alpha^2\left(\ln\alpha-\tfrac12\right)+C }[/math]

Setting α = 0 we have

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

Now one can deduce the identity above.

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

[math]\displaystyle{ K(n)=\frac{\bigl(\Gamma(n)\bigr)^{n-1}}{G(n)}. }[/math]

More prosaically, one may write

[math]\displaystyle{ K(n+1)=1^1 \cdot 2^2 \cdot 3^3 \cdots n^n. }[/math]

The first values are

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

External links

• Weisstein, Eric W.. "K-Function". http://mathworld.wolfram.com/K-Function.html.