Meijer G-function

The G- function was introduced by Cornelis Simon Meijer (1904-1974) in 1936. The most well-known special functions are special cases of this function.

There have been other approaches to generalize the special functions: the generalized hypergeometric function and the Macro Bertsche E- function have been proposed for the same purpose. The Meiersche G function includes these two functions as a special case. In its first definition Meijer uses a series. Today's traditional, more general definition is via a path integral in the complex plane (see definition below ), which was proposed by Arthur Erdélyi 1953. Most special functions can be shown closed with the aid of this information and of the gamma function.

By adding further parameters, the G- function for even more general Foxschen H function can be generalized ( introduced in 1961 by Charles Fox).

Definition

This path integral, along a suitable path in the complex plane can be regarded as an inverse Mellin transform. The integral exists under the following conditions:

• 0 ≤ m ≤ q, and 0 ≤ n ≤ P, wherein m, n, p and q are integers,
• Ak - bj ≠ 1, 2, 3, ..., (k = 1, 2, ..., n and j = 1, 2, ..., m). This ensures that no pole of Γ ( bj - s), j = 1, 2, ..., m with any pole of Γ (1 - ak s ), k = 1, 2, ..., n coincident
• .