View source for Meijer-G-functions

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

<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>

<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>

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" />:

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" />;

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" />;

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" />.

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).

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

<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>

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>

<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>

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,

<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>

<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>

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]]].

====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>