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.

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

$\zeta '(a,z)\ {\stackrel {\mathrm {def} }{=}}\ \left.{\frac {\partial \zeta (s,z)}{\partial s}}\right|_{s=a},\ \ \zeta (s,q)=\sum _{k=0}^{\infty }(k+q)^{-s}$

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 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 $\Delta f(x)=x\ln(x)$ 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 f such that:

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

Differentiating this identity now with respect to α 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 K-function we write

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

And so

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

Setting α = 0 we have

$\int _{0}^{1}\ln K(x)\,dx=\lim _{t\rightarrow 0}\left[{\tfrac {1}{2}}t^{2}\left(\ln t-{\tfrac {1}{2}}\right)\right]+C\ =C$

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

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

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

Similar to the multiplication formula for the gamma function:

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

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

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

External links

• Weisstein, Eric W. "K-Function". MathWorld.