K-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:

: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:

: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 \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.

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)

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)=\tfrac12\alpha^2\left(\ln\alpha-\tfrac12\right)+C

Setting we have

:\int_0^1 \ln K(x),dx=\lim_{t\rightarrow0}\left[\tfrac12 t^2\left(\ln t-\tfrac12\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

Similar to the multiplication formula for the gamma function: :\prod_{j=1}^{n-1}\Gamma\left(\frac jn \right) = \sqrt{\frac{(2\pi)^{n-1}}{n}}

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

: \prod_{j=1}^{n-1}K\left(\frac jn \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, ... .

References

References

1. Adamchik, Victor S.. (1998). "PolyGamma Functions of Negative Order". Journal of Computational and Applied Mathematics.
2. (2004). "A Generalized polygamma function". Integral Transforms and Special Functions.
3. (2024). "A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions: a Tutorial". Bitstream.
4. (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.