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K-function

From Wikipedia, the free encyclopedia

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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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]

Properties

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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

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

References

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  1. ^ Adamchik, Victor S. (1998), "PolyGamma Functions of Negative Order", Journal of Computational and Applied Mathematics, 100: 191–199, archived from the original on 2016-03-03
  2. ^ Espinosa, Olivier; Moll, Victor Hugo, "A Generalized polygamma function" (PDF), Integral Transforms and Special Functions, 15 (2): 101–115, archived (PDF) from the original on 2023-05-14
  3. ^ "A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions: a Tutorial" (PDF). Bitstream: 14. Archived (PDF) from the original on 2023-04-05.
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