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 the polygamma function is [1]
Or using the balanced generalization of the polygamma function: [2]
where A is the Glaisher constant.
Similar to the Bohr-Mollerup Theorem for the Gamma function, the log K-function is the unique (up to an additive constant) eventually 2-convex solution to the equation where is the forward difference operator. [3]
It can be shown that for α > 0:
This can be shown by defining a function f such that:
Differentiating this identity now with respect to α yields:
Applying the logarithm rule we get
By the definition of the K-function we write
And so
Setting α = 0 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
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