The Meijer G function is defined by the inverse Laplace transform

where

and  L is one of three types of integration paths , , and .

Contour  starts at  and finishes at .

Contour  starts at  and finishes at .

Contour  starts at  and finishes at .

All the paths , , and  put all  poles on the right and all other poles of the integrand (which must be of the form ) on the left.

The classical notation used to represent the MeijerG function relates to the notation used in Maple by

Note: See Prudnikov, Brychkov, and Marichev.

The MeijerG function satisfies the following th-order linear differential equation

where  and p is less than or equal to q.

Prudnikov, A. P.; Brychkov, Yu; and Marichev, O. Integrals and Series, Volume 3: More Special Functions. New York: Gordon and Breach Science Publishers, 1990.