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[1]

Or using balanced generalization of polygamma function:[2]

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

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

1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... ((sequence A002109 in the OEIS)).

ReferencesEdit

External linksEdit

  • Weisstein, Eric W. "K-Function". MathWorld.