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

Formally, the K-function is defined as

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

It can also be given in closed form as

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

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

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

Another expression using the polygamma function is ^[1]

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

Or using the 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.

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 ${\displaystyle \Delta f(x)=x\ln(x)}$ where ${\displaystyle \Delta }$ is the forward difference operator. ^[3]

Properties

It can be shown that for α > 0:

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

This can be shown by defining a function f such that:

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

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

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

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

${\displaystyle K(n)={\frac {{\bigl (}\Gamma (n){\bigr )}^{n-1}}{G(n)}}.}$

More prosaically, one may write

${\displaystyle K(n+1)=1^{1}\cdot 2^{2}\cdot 3^{3}\cdots n^{n}.}$

The first values are

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

Similar to the multiplication formula for the gamma function:

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

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

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

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 .