K-function

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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.

Formally, the K-function is defined as

K(z)=(2\pi)^{(-z+1)/2} \exp\left[\begin{pmatrix} z\\ 2\end{pmatrix}+\int_0^{z-1} \ln(t!)\,dt\right].

It can also be given in closed form as

K(z)=\exp\left[\zeta^\prime(-1,z)-\zeta^\prime(-1)\right]

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

\zeta^\prime(a,z)\ \stackrel{\mathrm{def}}{=}\ \left[\frac{d\zeta(s,z)}{ds}\right]_{s=a}.

Another expression using polygamma function is[1]

K(z)=\exp\left(\psi^{(-2)}(z)+\frac{z^2-z}{2}-\frac z2 \ln (2\pi)\right)

Or using balanced generalization of Polygamma function:[2]

K(z)=A e^{\psi(-2,z)+\frac{z^2-z}{2}}
where A is Glaisher constant.

The K-function is closely related to the Gamma function and the Barnes G-function; for natural numbers n, we have

K(n)=\frac{(\Gamma(n))^{n-1}}{G(n)}.

More prosaically, one may write

K(n+1)=1^1\, 2^2\, 3^3 \cdots n^n.

The first values are

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

References[edit]

  1. ^ Victor S. Adamchik. PolyGamma Functions of Negative Order
  2. ^ Olivier Espinosa Victor H. Moll. A Generalized polygamma function. Integral Transforms and Special Functions Vol. 15, No. 2, April 2004, pp. 101–115

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