Namespaces
Variants
Actions

Difference between revisions of "Meijer-G-functions"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (Undo revision 47819 by Ulf Rehmann (talk))
Tag: Undo
Line 1: Line 1:
−
<!--
+
Generalizations of the hypergeometric functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m1101102.png" /> of one variable (cf. also [[Hypergeometric function|Hypergeometric function]]). They can be defined by an integral as
−
m1101102.png
 
−
$#A+1 = 45 n = 0
 
−
$#C+1 = 45 : ~/encyclopedia/old_files/data/M110/M.1100110 Meijer \BMI G\EMI\AAhfunctions
 
−
Automatically converted into TeX, above some diagnostics.
 
−
Please remove this comment and the {{TEX|auto}} line below,
 
−
if TeX found to be correct.
 
−
-->
 
  
−
{{TEX|auto}}
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m1101103.png" /></td> </tr></table>
−
{{TEX|done}}
 
  
−
Generalizations of the hypergeometric functions  $  { {} _ {p} F _ {q} } $
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m1101104.png" /></td> </tr></table>
−
of one variable (cf. also [[Hypergeometric function|Hypergeometric function]]). They can be defined by an integral as
 
  
−
$$
+
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m1101105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m1101106.png" /> and the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m1101107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m1101108.png" /> are such that no pole of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m1101109.png" /> coincides with any pole of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011010.png" />. There are three possible choices for the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011011.png" />:
−
G _ {pq }  ^ {mn } \left ( x \left |
 
  
−
$$
+
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011012.png" /> goes from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011013.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011014.png" /> remaining to the right of the poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011015.png" /> and to the left of the poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011016.png" />;
−
=  
 
−
{
 
−
\frac{1}{2 \pi i }
 
−
} \int\limits _ { 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 ) }
 
−
} x  ^ {s} }  {ds } ,
 
−
$$
 
  
−
where  $  0 \leq  m \leq  p $,  
+
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011017.png" /> begins and ends at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011018.png" />, encircles counterclockwise all poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011019.png" /> and does not encircle any pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011020.png" />;
−
$  0 \leq  n \leq  q $
 
−
and the parameters  $  a _ {r} $,
 
−
$  b _ {r} $
 
−
are such that no pole of the functions  $  \Gamma ( b _ {j} - s ) $
 
−
coincides with any pole of the functions  $  \Gamma ( 1 - a _ {j} + s ) $.  
 
−
There are three possible choices for the contour  $  L $:
 
  
−
a) $  L $
+
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011021.png" /> begins and ends at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011022.png" />, encircles clockwise all poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011023.png" /> and does not encircle any pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011024.png" />.
−
goes from  $  - i \infty $
 
−
to  $  + i \infty $
 
−
remaining to the right of the poles of $  \Gamma ( b _ {j} - s ) $
 
−
and to the left of the poles of $  \Gamma ( 1 - a _ {j} + s ) $;
 
  
−
b) $  L $
+
The integral converges if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011026.png" /> in case a); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011027.png" /> and either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011028.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011030.png" /> in case b); and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011031.png" /> and either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011032.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011034.png" /> in case c).
−
begins and ends at  $  + \infty $,
 
−
encircles counterclockwise all poles of  $  \Gamma ( b _ {j} - s ) $
 
−
and does not encircle any pole of  $  \Gamma ( 1 - a _ {j} + s ) $;
 
  
−
c)  $  L $
+
The integral defining the Meijer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011035.png" />-functions can be calculated by means of the residue theorem and one obtains expressions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011036.png" /> in terms of the hypergeometric functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011037.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011038.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011039.png" /> satisfies the linear differential equation
−
begins and ends at  $  - \infty $,
 
−
encircles clockwise all poles of $  \Gamma ( 1 - a _ {j} + s ) $
 
−
and does not encircle any pole of $  \Gamma ( b _ {j} - s ) $.
 
  
−
The integral converges if  $  p + q < 2 ( m + n ) $,
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011040.png" /></td> </tr></table>
−
$  | { { \mathop{\rm arg} } x } | < ( m + n - {1 / 2 } ) ( p + q ) \pi $
 
−
in case a); if  $  q \geq  1 $
 
−
and either  $  p < q $
 
−
or  $  p = q $
 
−
and  $  | x | < 1 $
 
−
in case b); and if  $  p \geq  1 $
 
−
and either  $  p > q $
 
−
or  $  p = q $
 
−
and  $  | x | > 1 $
 
−
in case c).
 
−
 
 
−
The integral defining the Meijer  $  G $-
 
−
functions can be calculated by means of the residue theorem and one obtains expressions for  $  G _ {pq }  ^ {mn } $
 
−
in terms of the hypergeometric functions  $  { {} _ {p} F _ {q - 1 }  } $
 
−
or  $  { {} _ {q} F _ {p - 1 }  } $.
 
−
The function  $  G _ {pq }  ^ {mn } $
 
−
satisfies the linear differential equation
 
−
 
 
−
$$
 
−
{\mathcal L} y = 0,
 
−
$$
 
  
 
where
 
where
  
−
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011041.png" /></td> </tr></table>
−
{\mathcal L} =
 
−
$$
 
  
−
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011042.png" /></td> </tr></table>
−
=  
 
−
\left [ ( - 1 ) ^ {p - m - n } \prod _ {j = 1 } ^ { p }  \left ( x {
 
−
\frac{d}{dx }
 
−
} - a _ {j} + 1 \right ) \prod _ {j = 1 } ^ { q }  \left ( x {
 
−
\frac{d}{dx }
 
−
} - b _ {j} \right ) \right ] .
 
−
$$
 
  
−
Many functions of hypergeometric type and their products can be expressed in terms of Meijer $  G $-
+
Many functions of hypergeometric type and their products can be expressed in terms of Meijer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011043.png" />-functions, [[#References|[a1]]]. For example,
−
functions, [[#References|[a1]]]. For example,
 
  
−
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011044.png" /></td> </tr></table>
−
J _ {a - b }  ( 2 \sqrt x ) = x ^ {- ( a + b ) /2 } G _ {02 }  ^ {10 } ( x \mid  a,b ) ,
 
−
$$
 
  
−
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011045.png" /></td> </tr></table>
−
J _ {b - a }  ( \sqrt x ) Y _ {b - a }  ( \sqrt x ) = - \sqrt x x ^ {- a } G _ {13 }  ^ {20 } \left ( x \left |
 
  
−
Meijer $  G $-
+
Meijer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011046.png" />-functions appear in the theory of [[Lie group|Lie group]] representations (cf. also [[Representation of a compact group|Representation of a compact group]]) as transition coefficients for different bases of carrier spaces of representations [[#References|[a2]]].
−
functions appear in the theory of [[Lie group|Lie group]] representations (cf. also [[Representation of a compact group|Representation of a compact group]]) as transition coefficients for different bases of carrier spaces of representations [[#References|[a2]]].
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Erdelyi,  W. Magnus,  F. Oberhettinger,  F. Tricomi,  "Higher transcendental functions" , '''1''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.J. Vilenkin,  A.U. Klimyk,  "Representation of Lie groups and special functions" , '''2''' , Kluwer Acad. Publ.  (1993)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Erdelyi,  W. Magnus,  F. Oberhettinger,  F. Tricomi,  "Higher transcendental functions" , '''1''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.J. Vilenkin,  A.U. Klimyk,  "Representation of Lie groups and special functions" , '''2''' , Kluwer Acad. Publ.  (1993)  (In Russian)</TD></TR></table>

Revision as of 10:26, 7 June 2020

Generalizations of the hypergeometric functions of one variable (cf. also Hypergeometric function). They can be defined by an integral as

where , and the parameters , are such that no pole of the functions coincides with any pole of the functions . There are three possible choices for the contour :

a) goes from to remaining to the right of the poles of and to the left of the poles of ;

b) begins and ends at , encircles counterclockwise all poles of and does not encircle any pole of ;

c) begins and ends at , encircles clockwise all poles of and does not encircle any pole of .

The integral converges if , in case a); if and either or and in case b); and if and either or and in case c).

The integral defining the Meijer -functions can be calculated by means of the residue theorem and one obtains expressions for in terms of the hypergeometric functions or . The function satisfies the linear differential equation

where

Many functions of hypergeometric type and their products can be expressed in terms of Meijer -functions, [a1]. For example,

Meijer -functions appear in the theory of Lie group representations (cf. also Representation of a compact group) as transition coefficients for different bases of carrier spaces of representations [a2].

References

[a1] A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, "Higher transcendental functions" , 1 , McGraw-Hill (1953)
[a2] N.J. Vilenkin, A.U. Klimyk, "Representation of Lie groups and special functions" , 2 , Kluwer Acad. Publ. (1993) (In Russian)
How to Cite This Entry:
Meijer-G-functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meijer-G-functions&oldid=47819
This article was adapted from an original article by A.U. Klimyk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Meijer-G-functions&oldid=49295"