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.
Formally, the K-function is defined as
It can also be given in closed form as
where ζ'(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and
Another expression using polygamma function is
Or using balanced generalization of polygamma function:
- where A is Glaisher constant.
It can also be shown that for :
This can be shown by defining the function such that:
Deriving this identity now with respect to yields:
Applying the logarithm rule we get
By the definition of the K-Function we write
Setting we have
Now one can deduce the identity above.
The K-function is closely related to the gamma function and the Barnes G-function; for natural numbers n, we have
More prosaically, one may write
The first values are
- 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... ((sequence A002109 in the OEIS)).