K-function
In mathematics, the -function, typically denoted K, 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 Particular [values of the gamma function#Products|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