In mathematics, the -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 -function.
The direct definition:
:<math>K(z)=(2\pi)^{-\frac{z-1}2} \exp\left[\binom{z}{2}+\int_0^{z-1} \ln \Gamma(t + 1)\,dt\right].</math>
Definition via
:<math>K(z)=\exp\bigl[\zeta'(-1,z)-\zeta'(-1)\bigr]</math>
where denotes the derivative of the Riemann zeta function, denotes the Hurwitz zeta function and
:<math>\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}</math>
Definition via polygamma function:
:<math>K(z)=\exp\left[\psi^{(-2)}(z)+\frac{z^2-z}{2}-\frac {z}{2} \ln 2\pi \right]</math>
Definition via balanced generalization of the polygamma function:
:<math>K(z)=A \exp\left[\psi(-2,z)+\frac{z^2-z}{2}\right]</math>
where 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:<blockquote>Let <math>f: (0, \infty) \to \R</math> be a solution to the functional equation <math>f(x+1) - f(x)=x\ln x</math>, such that there exists some <math>M > 0 </math>, such that given any distinct <math>x_0, x_1, x_2, x_3 \in (M, \infty) </math>, the divided difference <math>f[x_0, x_1, x_2, x_3] \geq 0</math>.
Such functions are precisely <math>f = \ln K + C</math>, where <math>C</math> is an arbitrary constant.</blockquote>
Properties
For :
:<math>\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)</math>
Functional equations
The -function is closely related to the gamma function and the Barnes -function. For all complex <math>z</math>, <math display="block">K(z) G(z)=e^{(z-1) \ln \Gamma(z)}</math>
Multiplication formula
Similar to the multiplication formula for the gamma function:
:<math>\prod_{j=1}^{n-1}\Gamma\left(\frac jn \right) = \sqrt{\frac{(2\pi)^{n-1{n</math>
there exists a multiplication formula for the K-Function involving Glaisher's constant:
: <math>\prod_{j=1}^{n-1}K\left(\frac jn \right) = A^{\frac{n^2-1}{nn^{-\frac{1}{12ne^{\frac{1-n^2}{12n</math>
Integer values
For all non-negative integers,<math display="block">K(n+1)=1^1 \cdot 2^2 \cdot 3^3 \cdots n^n = H(n)</math>where <math>H</math> is the hyperfactorial.
The first values are
:1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... .