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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
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m1101102.png
 
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$#A+1 = 45 n = 0
 
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$#C+1 = 45 : ~/encyclopedia/old_files/data/M110/M.1100110 Meijer \BMI G\EMI\AAhfunctions
 
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<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>
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Generalizations of the hypergeometric functions  $  { {} _ {p} F _ {q} } $
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<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>
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of one variable (cf. also [[Hypergeometric function|Hypergeometric function]]). They can be defined by an integral as
 
  
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$$
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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" />:
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G _ {pq }  ^ {mn } \left ( x \left |
 
  
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$$
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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" />;
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=  
 
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{
 
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\frac{1}{2 \pi i }
 
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} \int\limits _ { L } { {
 
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\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 ) }
 
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} x  ^ {s} }  {ds } ,
 
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$$
 
  
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where  $  0 \leq  m \leq  p $,  
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b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011017.png" /> begins and ends at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011018.png" />, encircles counterclockwise all poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011019.png" /> and does not encircle any pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011020.png" />;
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$  0 \leq  n \leq  q $
 
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and the parameters  $  a _ {r} $,
 
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$  b _ {r} $
 
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are such that no pole of the functions  $  \Gamma ( b _ {j} - s ) $
 
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coincides with any pole of the functions  $  \Gamma ( 1 - a _ {j} + s ) $.  
 
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There are three possible choices for the contour  $  L $:
 
  
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a) $  L $
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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" />.
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goes from  $  - i \infty $
 
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to  $  + i \infty $
 
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remaining to the right of the poles of $  \Gamma ( b _ {j} - s ) $
 
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and to the left of the poles of $  \Gamma ( 1 - a _ {j} + s ) $;
 
  
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b) $  L $
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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).
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begins and ends at  $  + \infty $,
 
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encircles counterclockwise all poles of  $  \Gamma ( b _ {j} - s ) $
 
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and does not encircle any pole of  $  \Gamma ( 1 - a _ {j} + s ) $;
 
  
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c)  $  L $
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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
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begins and ends at  $  - \infty $,
 
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encircles clockwise all poles of $  \Gamma ( 1 - a _ {j} + s ) $
 
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and does not encircle any pole of $  \Gamma ( b _ {j} - s ) $.
 
  
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The integral converges if  $  p + q < 2 ( m + n ) $,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011040.png" /></td> </tr></table>
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$  | { { \mathop{\rm arg} } x } | < ( m + n - {1 / 2 } ) ( p + q ) \pi $
 
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in case a); if  $  q \geq  1 $
 
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and either  $  p < q $
 
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or  $  p = q $
 
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and  $  | x | < 1 $
 
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in case b); and if  $  p \geq  1 $
 
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and either  $  p > q $
 
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or  $  p = q $
 
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and  $  | x | > 1 $
 
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in case c).
 
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The integral defining the Meijer  $  G $-
 
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functions can be calculated by means of the residue theorem and one obtains expressions for  $  G _ {pq }  ^ {mn } $
 
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in terms of the hypergeometric functions  $  { {} _ {p} F _ {q - 1 }  } $
 
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or  $  { {} _ {q} F _ {p - 1 }  } $.
 
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The function  $  G _ {pq }  ^ {mn } $
 
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satisfies the linear differential equation
 
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$$
 
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{\mathcal L} y = 0,
 
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$$
 
  
 
where
 
where
  
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$$
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<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>
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{\mathcal L} =
 
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$$
 
  
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$$
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<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>
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=  
 
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\left [ ( - 1 ) ^ {p - m - n } \prod _ {j = 1 } ^ { p }  \left ( x {
 
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\frac{d}{dx }
 
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} - a _ {j} + 1 \right ) \prod _ {j = 1 } ^ { q }  \left ( x {
 
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\frac{d}{dx }
 
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} - b _ {j} \right ) \right ] .
 
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$$
 
  
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Many functions of hypergeometric type and their products can be expressed in terms of Meijer $  G $-
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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,
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functions, [[#References|[a1]]]. For example,
 
  
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$$
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<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>
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J _ {a - b }  ( 2 \sqrt x ) = x ^ {- ( a + b ) /2 } G _ {02 }  ^ {10 } ( x \mid  a,b ) ,
 
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$$
 
  
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$$
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<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>
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J _ {b - a }  ( \sqrt x ) Y _ {b - a }  ( \sqrt x ) = - \sqrt x x ^ {- a } G _ {13 }  ^ {20 } \left ( x \left |
 
  
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Meijer $  G $-
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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]]].
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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=49295
This article was adapted from an original article by A.U. Klimyk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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