The Meijer G function is traditionally defined by an inverse Mellin transform

Hence the traditional definition is related to our definition by

The new notation has some advantages over the old notation.  First, the parameters of the Meijer G function are separated out into four natural groups , , , and .  Second, possibly more controversial, we place instead of inside the integrand.  We deem this desirable because of the "multi-valued" character of .  Finally, the subscripts and superscripts which are now redundant are omitted.