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#edit

There are multiple equivalent definitions of the K-function.

The direct definition:

${\displaystyle K(z)=(2\pi )^{-{\frac {z-1}{2\exp \left[{\binom {z}{2+\int _{0}^{z-1}\ln \Gamma (t+1)\,dt\right].}$

Definition via

${\displaystyle K(z)=\exp {\bigl [}\zeta '(-1,z)-\zeta '(-1){\bigr ]$

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

${\displaystyle \zeta '(a,z)\ {\stackrel {\mathrm {def} }{=\ \left.{\frac {\partial \zeta (s,z)}{\partial s\right|_{s=a},\ \ \zeta (s,q)=\sum _{k=0}^{\infty }(k+q)^{-s$

Definition via polygamma function:^[1]

${\displaystyle K(z)=\exp \left[\psi ^{(-2)}(z)+{\frac {z^{2}-z}{2-{\frac {z}{2\ln 2\pi \right]}$

Definition via balanced generalization of the polygamma function:^[2]

${\displaystyle K(z)=A\exp \left[\psi (-2,z)+{\frac {z^{2}-z}{2\right]}$

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

Properties#edit

For α > 0:

${\displaystyle \int _{\alpha }^{\alpha +1}\ln K(x)\,dx-\int _{0}^{1}\ln K(x)\,dx={\tfrac {1}{2\alpha ^{2}\left(\ln \alpha -{\tfrac {1}{2\right)}$

Proof

Let ${\displaystyle f(\alpha )=\int _{\alpha }^{\alpha +1}\ln K(x)\,dx}$

Differentiating this identity now with respect to α yields:

${\displaystyle f'(\alpha )=\ln K(\alpha +1)-\ln K(\alpha )}$

Applying the logarithm rule we get

${\displaystyle f'(\alpha )=\ln {\frac {K(\alpha +1)}{K(\alpha )$

By the definition of the K-function we write

${\displaystyle f'(\alpha )=\alpha \ln \alpha }$

And so

${\displaystyle f(\alpha )={\tfrac {1}{2\alpha ^{2}\left(\ln \alpha -{\tfrac {1}{2\right)+C}$

Setting α = 0 we have

${\displaystyle \int _{0}^{1}\ln K(x)\,dx=\lim _{t\rightarrow 0}\left[{\tfrac {1}{2t^{2}\left(\ln t-{\tfrac {1}{2\right)\right]+C\ =C}$

Functional equations#edit

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

Multiplication formula#edit

${\displaystyle \prod _{j=1}^{n-1}\Gamma \left({\frac {j}{n\right)={\sqrt {\frac {(2\pi )^{n-1{n$

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

${\displaystyle \prod _{j=1}^{n-1}K\left({\frac {j}{n\right)=A^{\frac {n^{2}-1}{nn^{-{\frac {1}{12ne^{\frac {1-n^{2{12n$

Integer values#edit

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

The first values are

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

References#edit

^ 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#edit

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

Categories Gamma and related functions

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