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.
Definition
There are multiple equivalent definitions of the -function.
The direct definition:
Definition via
where denotes the derivative of the Riemann zeta function, denotes the Hurwitz zeta function and
Definition via polygamma function:
Definition via balanced generalization of the polygamma function:
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: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.
Properties
For :
Functional equations The -function is closely related to the gamma function and the Barnes -function. For all complex ,
Multiplication formula Similar to the multiplication formula for the gamma function:
there exists a multiplication formula for the K-Function involving Glaisher's constant:
Integer values For all non-negative integers,where is the hyperfactorial.
The first values are 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... . References
External links