K-function

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

There are multiple equivalent definitions of the $K$ -function.

The direct definition:

K (z) = (2 π)^{- \frac{z - 1}{2}} \exp [(\binom{z}{2}) + \int_{0}^{z - 1} \ln Γ (t + 1) d t] .

Definition via

K (z) = \exp [ζ^{'} (- 1, z) - ζ^{'} (- 1)]

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

ζ^{'} (a, z) \overset{d e f}{=} {\frac{\partial ζ (s, z)}{\partial s} |}_{s = a}, ζ (s, q) = \sum_{k = 0}^{\infty} (k + q)^{- s}

Definition via polygamma function:

K (z) = \exp [ψ^{(- 2)} (z) + \frac{z^{2} - z}{2} - \frac{z}{2} \ln 2 π]

Definition via balanced generalization of the polygamma function:

K (z) = A \exp [ψ (- 2, z) + \frac{z^{2} - z}{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 $f : (0, \infty) \to ℝ$ be a solution to the functional equation $f (x + 1) - f (x) = x \ln x$ , such that there exists some $M > 0$ , such that given any distinct $x_{0}, x_{1}, x_{2}, x_{3} \in (M, \infty)$ , the divided difference $f [x_{0}, x_{1}, x_{2}, x_{3}] \geq 0$ . Such functions are precisely $f = \ln K + C$ , where $C$ is an arbitrary constant.

Properties

For $α > 0$ :

\int_{α}^{α + 1} \ln K (x) d x - \int_{0}^{1} \ln K (x) d x = \frac{1}{2} α^{2} (\ln α - \frac{1}{2})

Proof

Proof

Let $f (α) = \int_{α}^{α + 1} \ln K (x) d x$

Differentiating this identity now with respect to $α$ yields:

f^{'} (α) = \ln K (α + 1) - \ln K (α)

Applying the logarithm rule we get

f^{'} (α) = \ln \frac{K (α + 1)}{K (α)}

By the definition of the $K$ -function we write

f^{'} (α) = α \ln α

And so

f (α) = \frac{1}{2} α^{2} (\ln α - \frac{1}{2}) + C

Setting $α = 0$ we have

\int_{0}^{1} \ln K (x) d x = lim_{t \to 0} [\frac{1}{2} t^{2} (\ln t - \frac{1}{2})] + C = C

Functional equations

The $K$ -function is closely related to the gamma function and the Barnes $G$ -function. For all complex $z$ , $K (z) G (z) = e^{(z - 1) \ln Γ (z)}$

Multiplication formula

\prod_{j = 1}^{n - 1} Γ (\frac{j}{n}) = \sqrt{\frac{(2 π)^{n - 1}}{n}}

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

\prod_{j = 1}^{n - 1} K (\frac{j}{n}) = A^{\frac{n^{2} - 1}{n}} n^{- \frac{1}{12 n}} e^{\frac{1 - n^{2}}{12 n}}

Integer values

For all non-negative integers, $K (n + 1) = 1^{1} \cdot 2^{2} \cdot 3^{3} \dots n^{n} = H (n)$ where $H$ is the hyperfactorial.

The first values are

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

References

↑ 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
↑ 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
↑ 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.
↑ 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.

External links

Weisstein, Eric W. "K-Function". MathWorld.

[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.

[:2-4] 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.