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.

Contents

Definition

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]

Properties

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 ).

References

  1. Adamchik, Victor S. (1998), "PolyGamma Functions of Negative Order", Journal of Computational and Applied Mathematics, 100 (2): 191–199, doi:10.1016/S0377-0427(98)00192-7, archived from the original on 2016-03-03
  2. Espinosa, Olivier; Moll, Victor Hugo (2004) [April 2004], "A Generalized polygamma function" (PDF), Integral Transforms and Special Functions, 15 (2): 101–115, doi:10.1080/10652460310001600573, archived (PDF) from the original on 2023-05-14
  3. Marichal, Jean-Luc; Zenaïdi, Naïm (2024). "A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions: a Tutorial" (PDF). Bitstream. 98 (2): 455–481. arXiv: 2207.12694 . doi:10.1007/s00010-023-00968-9. Archived (PDF) from the original on 2023-04-05.
  4. Sondow, Jonathan; Hadjicostas, Petros (2006-10-16). "The generalized-Euler-constant function γ(z) and a generalization of Somos's quadratic recurrence constant". Journal of Mathematical Analysis and Applications. 332: 292–314. arXiv: math/0610499 . doi:10.1016/j.jmaa.2006.09.081.