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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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<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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<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>
+
Generalizations of the hypergeometric functions  $  { {} _ {p} F _ {q} } $
 +
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" />;
+
$$
 +
=  
 +
{
 +
\frac{1}{2 \pi i }
 +
} \int\limits _ { L } { {
 +
\frac{\prod _ {j = 1 } ^ { m }  \Gamma ( b _ {j} - s ) \prod _ {j = 1 } ^ { n }  \Gamma ( 1 - a _ {j} + s ) }{\prod _ {j = m + 1 } ^ { q }  \Gamma ( 1 - b _ {j} + s ) \prod _ {j = n + 1 } ^ { p }  \Gamma ( a _ {j} - s ) }
 +
} x  ^ {s} }  {ds } ,
 +
$$
  
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" />;
+
where  $  0 \leq  m \leq  p $,
 +
$  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 $:
  
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" />.
+
a) $  L $
 +
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 ) $;
  
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).
+
b) $  L $
 +
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 ) $;
  
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
+
c)  $  L $
 +
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 ) $.
  
<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>
+
The integral converges if  $  p + q < 2 ( m + n ) $,
 +
$  | { { \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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110110/m11011043.png" />-functions, [[#References|[a1]]]. For example,
+
Many functions of hypergeometric type and their products can be expressed in terms of Meijer $  G $-
 +
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 <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]]].
+
Meijer $  G $-
 +
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 08:00, 6 June 2020


Generalizations of the hypergeometric functions $ { {} _ {p} F _ {q} } $ of one variable (cf. also Hypergeometric function). They can be defined by an integral as

$$ G _ {pq } ^ {mn } \left ( x \left | $$ = { \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 } ,

$$ where $ 0 \leq m \leq p $, $ 0 \leq n \leq q $ and the parameters $ a _ {r} $, $ b _ {r} $ are such that no pole of the functions $ \Gamma ( b _ {j} - s ) $ coincides with any pole of the functions $ \Gamma ( 1 - a _ {j} + s ) $. There are three possible choices for the contour $ L $: a) $ L $ goes from $ - i \infty $ to $ + i \infty $ remaining to the right of the poles of $ \Gamma ( b _ {j} - s ) $ and to the left of the poles of $ \Gamma ( 1 - a _ {j} + s ) $; b) $ L $ begins and ends at $ + \infty $, encircles counterclockwise all poles of $ \Gamma ( b _ {j} - s ) $ and does not encircle any pole of $ \Gamma ( 1 - a _ {j} + s ) $; c) $ L $ begins and ends at $ - \infty $, encircles clockwise all poles of $ \Gamma ( 1 - a _ {j} + s ) $ and does not encircle any pole of $ \Gamma ( b _ {j} - s ) $. The integral converges if $ p + q < 2 ( m + n ) $, $ | { { \mathop{\rm arg} } x } | < ( m + n - {1 / 2 } ) ( p + q ) \pi $ in case a); if $ q \geq 1 $ and either $ p < q $ or $ p = q $ and $ | x | < 1 $ in case b); and if $ p \geq 1 $ and either $ p > q $ or $ p = q $ and $ | x | > 1 $ in case c). The integral defining the Meijer $ G $- functions can be calculated by means of the residue theorem and one obtains expressions for $ G _ {pq } ^ {mn } $ in terms of the hypergeometric functions $ { {} _ {p} F _ {q - 1 } } $ or $ { {} _ {q} F _ {p - 1 } } $. The function $ G _ {pq } ^ {mn } $ satisfies the linear differential equation $$ {\mathcal L} y = 0, $$ where $$ {\mathcal L} = $$ $$ = \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 ] .

$$ Many functions of hypergeometric type and their products can be expressed in terms of Meijer $ G $- functions, [[#References|[a1]]]. For example, $$ J _ {a - b } ( 2 \sqrt x ) = x ^ {- ( a + b ) /2 } G _ {02 } ^ {10 } ( x \mid a,b ) , $$ $$ J _ {b - a } ( \sqrt x ) Y _ {b - a } ( \sqrt x ) = - \sqrt x x ^ {- a } G _ {13 } ^ {20 } \left ( x \left |

Meijer $ G $- 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