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: be a solution to the functional equation, the divided difference.

is an arbitrary constant.

Properties

For :

Functional equations

The -function is closely related to the gamma function and the Barnes G-function

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

is the hyperfactorial.

The first values are

1. 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... .

References

External links