# Meijer-G-functions

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. A.U. Klimyk (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Meijer-G-functions&oldid=13688