K-function

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]