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.
There are multiple equivalent definitions of the K-function.
The direct definition:
Definition via
where ζ′(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and
Definition via polygamma function: [1]
Definitio via balanced generalization of the polygamma function: [2]
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 be a solution to the functional equation , such that there exists some , such that given any distinct , the divided difference . Such functions are precisely , where is an arbitrary constant. [3]
For α > 0:
Let
Differentiating this identity now with respect to α yields:
Applying the logarithm rule we get
By the definition of the K-function we write
And so
Setting α = 0 we have
The K-function is closely related to the gamma function and the Barnes G-function. For all complex ,
Similar to the multiplication formula for the gamma function:
there exists a multiplication formula for the K-Function involving Glaisher's constant: [4]
For all non-negative integers,where is the hyperfactorial.
The first values are