K-function

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:

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

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