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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.

Definition

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There are multiple equivalent definitions of the K-function.

The direct definition:

Definition via

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

Definition via polygamma function:[1]

Definitio via balanced generalization of the polygamma function:[2]

where A is the Glaisher constant.

It can be defined via unique characterization, similar to how the gamma function can be uniquely characterized by the Bohr-Mollerup Theorem:

Let be a solution to the functional equation , such that there exists some , such that given any distinct , the divided difference . Such functions are precisely , where is an arbitrary constant.[3]

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Properties

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For α > 0:

Proof
Proof

Let

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

Functional equations

The K-function is closely related to the gamma function and the Barnes G-function. For all complex ,

Multiplication formula

Similar to the multiplication formula for the gamma function:

there exists a multiplication formula for the K-Function involving Glaisher's constant:[4]

Integer values

For all non-negative integers,where is the hyperfactorial.

The first values are

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

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