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.

There are multiple equivalent definitions of the K-function.

The direct definition:

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

Definition via

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

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

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

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

For α > 0:

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

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

Similar to the multiplication formula for the gamma function:

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

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

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

The first values are

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

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

• Gamma and related functions