K-function
In mathematics, the K-function, typically denoted K, 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' denotes the derivative of the Riemann zeta function, ζ denotes the Hurwitz zeta function and
Another expression using polygamma function is
Or using balanced generalization of polygamma function:
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
And so
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
