# Meijer-G-functions

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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].

How to Cite This Entry:
Meijer-G-functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meijer-G-functions&oldid=13688
This article was adapted from an original article by A.U. Klimyk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article