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:
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:<blockquote>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.</blockquote>
For :
The -function is closely related to the gamma function and the Barnes -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:
For all non-negative integers,where is the hyperfactorial.
The first values are