K-function

Formally, the {{mvar|K}}-function is defined as

:$K(z)=(2\backslash pi)^\{-\backslash frac\{z-1\}2\}\; \backslash exp\backslash left[\backslash binom\{z\}\{2\}+\backslash int\_0^\{z-1\}\; \backslash ln\; \backslash Gamma(t\; +\; 1)\backslash ,dt\backslash right].$

It can also be given in closed form as

:$K(z)=\backslash exp\backslash bigl[\backslash zeta\text{'}(-1,z)-\backslash zeta\text{'}(-1)\backslash bigr]$

where {{math|ζ′(z)}} denotes the derivative of the Riemann zeta function, {{math|ζ(a,z)}} denotes the Hurwitz zeta function and

:$\backslash zeta\text{'}(a,z)\backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash left.\backslash frac\{\backslash partial\backslash zeta(s,z)\}\{\backslash partial\; s\}\backslash right|\_\{s=a\},\backslash \; \backslash \; \backslash zeta(s,q)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; (k+q)^\{-s\}$

Another expression using the polygamma function is{{Citation| url =https://www.cs.cmu.edu/~adamchik/articles/polyg.htm | first=Victor S. | last =Adamchik | title =PolyGamma Functions of Negative Order | url-status =dead | archive-url =https://web.archive.org/web/20160303165448/http://www.cs.cmu.edu/~adamchik/articles/polyg.htm | archive-date=2016-03-03 | journal =Journal of Computational and Applied Mathematics | volume =100 | year=1998 | issue=2 | pages =191–199| doi=10.1016/S0377-0427(98)00192-7 | url-access =subscription }}

:$K(z)=\backslash exp\backslash left[\backslash psi^\{(-2)\}(z)+\backslash frac\{z^2-z\}\{2\}-\backslash frac\; \{z\}\{2\}\; \backslash ln\; 2\backslash pi\; \backslash right]$

Or using the balanced generalization of the polygamma function:{{Citation| url =http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf | title=A Generalized polygamma function | archive-url =https://web.archive.org/web/20230514164433/http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf | archive-date =2023-05-14 | first1 =Olivier | last1 =Espinosa | first2 =Victor Hugo | last2 =Moll | author-link2 =Victor Moll | journal =Integral Transforms and Special Functions | date=2004 | volume =15 | issue =2 | orig-date =April 2004 | pages =101–115 | doi=10.1080/10652460310001600573 | url-status =live}}

:$K(z)=A\; \backslash exp\backslash left[\backslash psi(-2,z)+\backslash frac\{z^2-z\}\{2\}\backslash right]$

where {{mvar|A}} is the Glaisher constant.

Similar to the Bohr-Mollerup Theorem for the gamma function, the log K-function is the unique (up to an additive constant) eventually 2-convex solution to the equation $\backslash Delta\; f(x)=x\backslash ln(x)$ where $\backslash Delta$ is the forward difference operator.{{Cite journal | title=A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions: a Tutorial | journal=Bitstream| url=https://orbilu.uni.lu/bitstream/10993/51793/1/AGeneralizationOfBohrMollerupTutorial.pdf |pages=455–481 | archive-url =https://web.archive.org/web/20230405060849/https://orbilu.uni.lu/bitstream/10993/51793/1/AGeneralizationOfBohrMollerupTutorial.pdf | archive-date =2023-04-05 | url-status =live| last1=Marichal| first1=Jean-Luc| last2=Zenaïdi| first2=Naïm| date=2024| volume=98| issue=2| doi=10.1007/s00010-023-00968-9| arxiv=2207.12694}}

It can be shown that for {{math|α > 0}}:

:$\backslash int\_\backslash alpha^\{\backslash alpha+1\}\backslash ln\; K(x)\backslash ,dx-\backslash int\_0^1\backslash ln\; K(x)\backslash ,dx=\backslash tfrac\{1\}\{2\}\backslash alpha^2\backslash left(\backslash ln\backslash alpha-\backslash tfrac\{1\}\{2\}\backslash right)$

This can be shown by defining a function {{mvar|f}} such that:

:$f(\backslash alpha)=\backslash int\_\backslash alpha^\{\backslash alpha+1\}\backslash ln\; K(x)\backslash ,dx$

Differentiating this identity now with respect to {{mvar|α}} yields:

:$f\text{'}(\backslash alpha)=\backslash ln\; K(\backslash alpha+1)-\backslash ln\; K(\backslash alpha)$

Applying the logarithm rule we get

:$f\text{'}(\backslash alpha)=\backslash ln\backslash frac\{K(\backslash alpha+1)\}\{K(\backslash alpha)\}$

By the definition of the {{mvar|K}}-function we write

:$f\text{'}(\backslash alpha)=\backslash alpha\backslash ln\backslash alpha$

And so

:$f(\backslash alpha)=\backslash tfrac12\backslash alpha^2\backslash left(\backslash ln\backslash alpha-\backslash tfrac12\backslash right)+C$

Setting {{math|α {{=}} 0}} we have

:$\backslash int\_0^1\; \backslash ln\; K(x)\backslash ,dx=\backslash lim\_\{t\backslash rightarrow0\}\backslash left[\backslash tfrac12\; t^2\backslash left(\backslash ln\; t-\backslash tfrac12\backslash right)\backslash right]+C\; \backslash \; =C$

Now one can deduce the identity above.

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

:$K(n)=\backslash frac\{\backslash bigl(\backslash Gamma(n)\backslash bigr)^\{n-1\}\}\{G(n)\}.$

More prosaically, one may write

:$K(n+1)=1^1\; \backslash cdot\; 2^2\; \backslash cdot\; 3^3\; \backslash cdots\; n^n.$

The first values are

:1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... {{OEIS|A002109}}.

Similar to the multiplication formula for the gamma function:

:$\backslash prod\_\{j=1\}^\{n-1\}\backslash Gamma\backslash left(\backslash frac\; jn\; \backslash right)\; =\; (2\; \backslash pi)^\{\backslash frac\{n-1\}\{2\}\}\; n^\{-\backslash frac\{n\}\{2\}\}$

there exists a multiplication formula for the K-Function involving Glaisher's constant:{{Cite journal |last1=Sondow |first1=Jonathan |last2=Hadjicostas |first2=Petros |date=2006-10-16 |title=The generalized-Euler-constant function γ(z) and a generalization of Somos's quadratic recurrence constant |journal=Journal of Mathematical Analysis and Applications |language=en |volume=332 |pages=292–314 |arxiv=math/0610499 |doi=10.1016/j.jmaa.2006.09.081}}

: $\backslash prod\_\{j=1\}^\{n-1\}K\backslash left(\backslash frac\; jn\; \backslash right)\; =\; A^\{\backslash frac\{n^2-1\}\{n\}\}n^\{-\backslash frac\{1\}\{12n\}\}e^\{\backslash frac\{1-n^2\}\{12n\}\}$

• {{mathworld|title=K-Function|urlname=K-Function}}

Category:Gamma and related functions