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