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.

Formally, the K-function is defined as

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

It can also be given in closed form as

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

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

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

Another expression using polygamma function is[1]

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

Or using balanced generalization of polygamma function:[2]

${\displaystyle K(z)=Ae^{\psi (-2,z)+{\frac {z^{2}-z}{2}}}}$

where A is Glaisher constant.

It can also be shown that for ${\displaystyle \alpha >0}$:

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

This can be shown by defining the function ${\displaystyle f}$ such that:

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

Deriving this identity now with respect to ${\displaystyle \alpha }$ yields:

${\displaystyle f'(\alpha )=\ln(K(\alpha +1))-\ln(K(\alpha ))}$

Applying the logarithm rule we get

${\displaystyle f'(\alpha )=\ln \left({\frac {K(\alpha +1)}{K(\alpha )}}\right)}$

By the definition of the K-Function we write

${\displaystyle f'(\alpha )=\alpha \ln(\alpha )}$

And so

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

Setting ${\displaystyle \alpha =0}$ we have

${\displaystyle \int _{0}^{1}\ln(K(x))dx=\lim _{n\rightarrow 0}\left({\frac {1}{2}}n^{2}\left(\ln(n)-{\frac {1}{2}}\right)\right)+C}$

${\displaystyle \int _{0}^{1}\ln(K(x))dx=C}$

Now one can deduce the identity above.

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

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

More prosaically, one may write

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

The first values are

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

References

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

External links

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