K function

The function is a special function in mathematics that is usually referred to as. She generalizes the hyper-faculty to the complex numbers ; analogous to the complex expansion of the factorial function to the gamma function . ${\displaystyle K}$ ${\displaystyle K(z)}$ ${\displaystyle H(n)}$

The hyperfaculty of a natural number is defined by ${\displaystyle n}$

${\displaystyle H(n)=\prod _{i=1}^{n}i^{i}=1^{1}2^{2}3^{3}4^{4}\cdots n^{n},\qquad n\in \mathbb {N} .}$

The function should now apply ${\displaystyle K}$

${\displaystyle K(n+1)=H(n),\qquad n\in \mathbb {N} ,}$

and it should be extended to the range of complex numbers.

Definitions

One possible definition of the function is: ${\displaystyle K}$

${\displaystyle K(z)=(2\pi )^{(-z+1)/2}\exp \left[{\binom {z}{2}}+\int \limits _{0}^{z-1}\ln(\Gamma (t+1))\;\mathrm {d} t\right],}$

where for the complex generalization of the binomial coefficient and Γ for the gamma function . ${\displaystyle {\tbinom {z}{2}}}$

Offers another option

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

where stand for the Riemann zeta function and for the Hurwitz zeta function (the derivatives are used in each case.) ${\displaystyle \zeta (z)}$ ${\displaystyle \zeta (a,z)}$

The relationship of the function to the gamma function and the Barnesian function is shown by the formula ${\displaystyle K}$ ${\displaystyle G}$

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

expressed.

values

For natural, the values ​​of the K-function coincide by definition with the value of the hyperfaculty function . The first of these values ​​are ${\displaystyle n}$ ${\displaystyle K(n)}$ ${\displaystyle H(n-1)}$

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

The value is given explicitly by ${\displaystyle K({\tfrac {1}{2}})}$

${\displaystyle K({\tfrac {1}{2}})={\frac {A^{3/2}}{2^{1/24}\cdot e^{1/8}}}}$ = 1.2451432494 ...

where stands for the constant of Glaisher-Kinkelin . ${\displaystyle A}$

Other connections

With the Barnesian G function, the ${\displaystyle G(z)}$ following applies

${\displaystyle K(z)\cdot G(z)=\exp \left\{(z-1)\cdot \log[\Gamma (z)]\right\},}$

for all ${\displaystyle z\in \mathbb {C} .}$

In 2003, Benoit Cloitre showed the following formula:

${\displaystyle {\frac {1}{K(n)}}=(-1)^{n}\operatorname {det} {\begin{vmatrix}-1&-1&-1&\cdots &-1\\{\frac {1}{2}}&{\frac {1}{4}}&{\frac {1}{8}}&\cdots &{\frac {1}{2^{n}}}\\-{\frac {1}{3}}&-{\frac {1}{9}}&-{\frac {1}{27}}&\cdots &-{\frac {1}{3^{n}}}\\\vdots &\vdots &\vdots &\ddots &\vdots \\{\frac {(-1)^{n}}{n}}&{\frac {(-1)^{n}}{n^{2}}}&{\frac {(-1)^{n}}{n^{3}}}&\cdots &{\frac {(-1)^{n}}{n^{n}}}\\\end{vmatrix}}}$.

Individual evidence

1. ↑ ^{a b c} Eric W. Weisstein : Hyperfactorial . In: MathWorld (English).
2. ↑ http://www.wolframalpha.com/input/?i=K-Function(1/2)

literature

• Hermann Kinkelin : About a transcendent related to the gamma function and its application to the integral calculus , Journal for pure and applied mathematics 57, 1860, 18, pp. 122-138 ( online )

Web links

• Eric W. Weisstein : K-function . In: MathWorld (English).