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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

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

The direct definition:

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

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

\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]

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]

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 $f:(0,\infty )\to \mathbb {R}$ 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.^[3]

Properties

For $α > 0$ :

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

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

f'(\alpha )=\ln K(\alpha +1)-\ln K(\alpha )

Applying the logarithm rule we get

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

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

f'(\alpha )=\alpha \ln \alpha

And so

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

Setting $α = 0$ we have

\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 $z$ , $K(z)G(z)=e^{(z-1)\ln \Gamma (z)}$

Multiplication formula

\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]

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

[rdp-we-cite_note-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

[rdp-we-cite_note-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

[rdp-we-cite_note-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.

[rdp-we-cite_note-: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.

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