Barnesian G function

The Barnesian function ${\displaystyle G}$ , typically denoted by, is a function that represents an extension of the super faculties to the complex numbers . It is related to the gamma function , the function and the Glaisher-Kinkelin constants and is named after the mathematician Ernest William Barnes . ${\displaystyle G(z)}$ ${\displaystyle K}$

Formally, the Barnesian function is defined in the form of a Weierstrass product as ${\displaystyle G}$

${\displaystyle G(z+1)=(2\pi )^{z/2}e^{-[z(z+1)+\gamma z^{2}]/2}\prod _{n=1}^{\infty }\left[\left(1+{\frac {z}{n}}\right)^{n}e^{-z+z^{2}/(2n)}\right]}$

where denotes the Euler-Mascheroni constant . ${\displaystyle \gamma }$

Difference equation, functional equation and special values

The Barnesian function satisfies the difference equation ${\displaystyle G}$

${\displaystyle \!\ G(z+1)=\Gamma (z)G(z)}$

with normalization The difference equation implies that takes the following values ​​for integer arguments: ${\displaystyle G(1)=1.}$ ${\displaystyle G}$

${\displaystyle G(n)={\begin{cases}0,&{\mbox{falls }}n=0,-1,-2,\ldots ,\\\prod _{i=0}^{n-2}i!,&{\mbox{falls }}n=1,2,\ldots ,\end{cases}}}$

so that

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

where denote the gamma function and the K function . The difference equation uniquely defines the function if the convexity condition is set. ${\displaystyle \Gamma (n)}$ ${\displaystyle K(n)}$ ${\displaystyle G}$ ${\displaystyle {\frac {\mathrm {d} ^{3}}{\mathrm {d} x^{3}}}G(x)\geq 0}$

The difference equation of the -function and the functional equation of the gamma function provide the following functional equation for the -function, as originally proven by Hermann Kinkelin : ${\displaystyle G}$ ${\displaystyle G}$

${\displaystyle G(1-z)=G(1+z){\frac {1}{(2\pi )^{z}}}\exp \int \limits _{0}^{z}\pi t\cot \pi t\,\mathrm {d} t.}$

Multiplication formula

Like the gamma function, the function also fulfills a multiplication formula: ${\displaystyle G}$

${\displaystyle G(nz)=K(n)n^{n^{2}z^{2}/2-nz}(2\pi )^{-{\frac {n^{2}-n}{2}}z}\prod _{i=0}^{n-1}\prod _{j=0}^{n-1}G\left(z+{\frac {i+j}{n}}\right)}$

where is a function performed by ${\displaystyle K(n)}$

${\displaystyle K(n)=e^{-(n^{2}-1)\zeta ^{\prime }(-1)}\cdot n^{\frac {5}{12}}\cdot (2\pi )^{(n-1)/2}\,=\,(Ae^{-{\frac {1}{12}}})^{n^{2}-1}\cdot n^{\frac {5}{12}}\cdot (2\pi )^{(n-1)/2}.}$

given is. Here is the derivative of the Riemann zeta function and the constant from Glaisher-Kinkelin . ${\displaystyle \zeta ^{\prime }}$ ${\displaystyle A}$

Asymptotic development

The function has the following asymptotic expansion found by Barnes: ${\displaystyle \log \,G(z+1)}$

${\displaystyle \log G(z+1)={\frac {1}{12}}-\log A+{\frac {z}{2}}\log 2\pi +\left({\frac {z^{2}}{2}}-{\frac {1}{12}}\right)\log z-{\frac {3z^{2}}{4}}+\sum _{k=1}^{N}{\frac {B_{2k+2}}{4k\left(k+1\right)z^{2k}}}+O\left({\frac {1}{z^{2N+2}}}\right).}$

Here denotes the Bernoulli numbers and the Glaisher-Kinkelin constant . (Note that at Barnes' time the Bernoulli number was written as . This convention is no longer used.) The expansion is valid for in any sector that does not contain the negative real axis. ${\displaystyle B_{k}}$ ${\displaystyle A}$ ${\displaystyle B_{2k}}$ ${\displaystyle (-1)^{k+1}B_{k}}$ ${\displaystyle z}$

Web link

• Eric W. Weisstein : Barnes Function ${\displaystyle G}$ . In: MathWorld (English).

Individual evidence

1. ^ Ernest W. Barnes: The theory of the function ${\displaystyle G}$ . In: The Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pages 264-314.
2. ^ Marie-France Vignéras : L'équation fonctionelle de la fonction zêta de Selberg du groupe modulaire ${\displaystyle SL(2,\mathbb {Z} )}$ . In: Astérisque , Vol. 61 (1979), pp. 235-249, ISSN  0303-1179 .
3. ^ Moshe Y. Vardi : Determinants of Laplacians and multiple gamma functions. In: SIAM Journal on Mathematical Analysis , Vol. 19 (1988), pages 493-507, ISSN  0036-1410 .
4. ^ Edmund Taylor Whittaker , George N. Watson: A Course of Modern Analysis. 4th ed. Cambridge University Press, Cambridge 1990, ISBN 978-0-521-09189-3 .