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.

For the k-function, see Bateman function.

Definition

There are multiple equivalent definitions of the K-function.

The direct definition:

${\displaystyle K(z)=(2\pi )^{-{\frac {z-1}{2}}}\exp \left[{\binom {z}{2}}+\int _{0}^{z-1}\ln \Gamma (t+1)\,dt\right].}$

Definition via

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

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

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

Definition via polygamma function:^[1]

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

Definition via balanced generalization of the polygamma function:^[2]

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

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 ${\displaystyle f:(0,\infty )\to \mathbb {R} }$ be a solution to the functional equation ${\displaystyle f(x+1)-f(x)=x\ln x}$, such that there exists some ${\displaystyle M>0}$, such that given any distinct ${\displaystyle x_{0},x_{1},x_{2},x_{3}\in (M,\infty )}$, the divided difference ${\displaystyle f[x_{0},x_{1},x_{2},x_{3}]\geq 0}$. Such functions are precisely ${\displaystyle f=\ln K+C}$, where ${\displaystyle C}$ is an arbitrary constant.^[3]

Properties

For α > 0:

${\displaystyle \int _{\alpha }^{\alpha +1}\ln K(x)\,dx-\int _{0}^{1}\ln K(x)\,dx={\tfrac {1}{2}}\alpha ^{2}\left(\ln \alpha -{\tfrac {1}{2}}\right)}$

Proof

Proof

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

Differentiating this identity now with respect to α yields:

${\displaystyle f'(\alpha )=\ln K(\alpha +1)-\ln K(\alpha )}$

Applying the logarithm rule we get

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

By the definition of the K-function we write

${\displaystyle f'(\alpha )=\alpha \ln \alpha }$

And so

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

Setting α = 0 we have

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

Functional equations

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

Multiplication formula

Similar to the multiplication formula for the gamma function:

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

there exists a multiplication formula for the K-Function involving Glaisher's constant:^[4]

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

Integer values

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

The first values are

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