Meijer G function

The G-function was introduced by Cornelis Simon Meijer (1904–1974) in 1936. Most of the known special functions are special cases of this function .

There have also been other approaches to generalize the special functions: the generalized hypergeometric function and the MacRobert E function have been proposed for the same purpose. The Meiersche G function comprises these two functions as a special case. In its first definition, Meijer uses a number. The more general definition that is customary today is based on a path integral in the complex number plane (see definition below), which was proposed by Arthur Erdélyi in 1953. With the help of this definition and the gamma function , most of the special functions can be displayed in closed form.

By adding further parameters, the G-function can be generalized to the even more general Fox's H-function (introduced in 1961 by Charles Fox ).

definition

${\displaystyle G_{p,q}^{\,m,n}\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\;z\right)={\frac {1}{2\pi i}}\int \limits _{\mathcal {L}}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}-s)\prod _{j=1}^{n}\Gamma (1-a_{j}+s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}+s)\prod _{j=n+1}^{p}\Gamma (a_{j}-s)}}z^{s}\,\mathrm {d} s}$,

where is the gamma function . This path integral along a suitable path in the complex number plane can be understood as an inverse Mellin transformation . The integral exists under the following conditions: ${\displaystyle \Gamma (\cdot )}$ ${\displaystyle {\mathcal {L}}}$

• ${\displaystyle 0\leq m\leq q}$and , where and are whole numbers , ${\displaystyle 0\leq n\leq p}$ ${\displaystyle m,n,p}$ ${\displaystyle q}$
• ${\displaystyle a_{k}-b_{j}\neq 1,2,3,\ldots }$( and ). This ensures that no pole of , with any pole of , coincides, ${\displaystyle k=1,2,\ldots ,n}$ ${\displaystyle j=1,2,\ldots ,m}$ ${\displaystyle \Gamma (b_{j}-s)}$ ${\displaystyle j=1,2,\ldots ,m}$ ${\displaystyle \Gamma (1-a_{k}+s)}$ ${\displaystyle k=1,2,\ldots ,n}$
• ${\displaystyle z\neq 0}$.

literature

• Larry C. Andrews: Special Functions of mathematics for Engineers , New York, ISBN 0-8194-2616-4 .
• Cornelis Simon Meijer: About Whittakersche respectively. Bessel functions and their products , Nieuw Archief voor Wiskunde 18 (4), 1936.

Web links

• Meijersche G-function on Wolfram-Mathworld (with graphic representations)
• Meijersche G-function on Wolfram-Mathworld (systematic)
• Richard Beals, Jacek Szmigielski: Meijer G-functions: a gentle introduction , Notices AMS, August 2013.

Individual evidence

1. ↑ Definition of the G function on Wolfram Mathworld