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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


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].


[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. A.U. Klimyk (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098