K-function

For the k-function, see Bateman 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:

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

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]

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

Definitio 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

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

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

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

Functional equations

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

Similar to the multiplication formula for the gamma function:

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

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

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

Integer values

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

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.

External links

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