K-function

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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

[K(z)=(2\pi)^ \exp\left[begin z\ 2end+int_0^ ln(t!)dtright].]

It can also be given in closed form as

[K(z)=\exp\left[zeta^prime(-1,z)-zeta^prime(-1)right]]

where ζ'(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and

[\zeta^\prime(a,z)\equiv\left[fracright]_.]

The K-function is closely related to the Gamma function and the Barnes G-function; for natural numbers n, we have

[K(n)=\frac}.]

More prosaically, one may write

[K(n+1)=1^1\, 2^2\, 3^3 \ldots n^n.]

References

, [K-Function] at MathWorld.

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