In mathematics, the -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.
There are multiple equivalent definitions of the -function.
The direct definition:
| ||||
| K(z)=(2\pi) |
| z-1 | |
| \exp\left[\binom{z}{2}+\int | |
| 0 |
ln\Gamma(t+1)dt\right].
Definition via
K(z)=\expl[\zeta'(-1,z)-\zeta'(-1)r]
where denotes the derivative of the Riemann zeta function, denotes the Hurwitz zeta function and
\zeta'(a,z) \stackrel{def
Definition via polygamma function:
K(z)=\exp\left[\psi(-2)(z)+
| z2-z | - | |
| 2 |
| z | |
| 2 |
ln2\pi\right]
Definition via balanced generalization of the polygamma function:
K(z)=A\exp\left[\psi(-2,z)+
| z2-z | |
| 2 |
\right]
where 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:
Letbe a solution to the functional equationf:(0,infty)\to\R
, such that there exists somef(x+1)-f(x)=xlnx
, such that given any distinctM>0
, the divided differencex0,x1,x2,x3\in(M,infty)
.f[x0,x1,x2,x3]\geq0
Such functions are precisely
, wheref=lnK+C
is an arbitrary constant.[1]C
For :
| \alpha+1 | |
| \int | |
| \alpha |
ln
| 1ln | |
| K(x)dx-\int | |
| 0 |
K(x)dx=\tfrac{1}{2}\alpha2\left(ln\alpha-\tfrac{1}{2}\right)
The -function is closely related to the gamma function and the Barnes -function. For all complex
z
Similar to the multiplication formula for the gamma function:
| n-1 | ||
| \prod | \Gamma\left( | |
| j=1 |
| jn | |
| \right) |
=\sqrt{
| (2\pi)n-1 | |
| n |
there exists a multiplication formula for the K-Function involving Glaisher's constant:[2]
| n-1 | ||
| \prod | K\left( | |
| j=1 |
| jn | |
| \right) |
=
| ||||
| A |
| ||||
| n |
| ||||
| e |
For all non-negative integers,where
H
The first values are
1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... .
