Meijer G-function - Definition of The Meijer G-function

Definition of The Meijer G-function

A general definition of the Meijer G-function is given by the following line integral in the complex plane (Bateman & Erdélyi 1953, § 5.3-1):

where Γ denotes the gamma function. This integral is of the so-called Mellin–Barnes type, and may be viewed as an inverse Mellin transform. The definition holds under the following assumptions:

• 0 ≤ m ≤ q and 0 ≤ n ≤ p, where m, n, p and q are integer numbers
• a_k − b_j ≠ 1, 2, 3, ... for k = 1, 2, ..., n and j = 1, 2, ..., m, which implies that no pole of any Γ(b_j − s), j = 1, 2, ..., m, coincides with any pole of any Γ(1 − a_k + s), k = 1, 2, ..., n
• z ≠ 0

Note that for historical reasons the first lower and second upper index refer to the top parameter row, while the second lower and first upper index refer to the bottom parameter row. One often encounters the following more synthetic notation using vectors:

Implementions of the G-function in computer algebra systems typically employ separate vector arguments for the four (possibly empty) parameter groups a₁ ... a_n, a_n+1 ... a_p, b₁ ... b_m, and b_m+1 ... b_q, and thus can omit the orders p, q, n, and m as redundant.

The L in the integral represents the path to be followed while integrating. Three choices are possible for this path:

1. L runs from −i∞ to +i∞ such that all poles of Γ(b_j − s), j = 1, 2, ..., m, are on the right of the path, while all poles of Γ(1 − a_k + s), k = 1, 2, ..., n, are on the left. The integral then converges for |arg z| < δ π, where

an obvious prerequisite for this is δ > 0. The integral additionally converges for |arg z| = δ π ≥ 0 if (q − p) (σ + 1⁄₂) > Re(ν) + 1, where σ represents Re(s) as the integration variable s approaches both +i∞ and −i∞, and where

As a corollary, for |arg z| = δ π and p = q the integral converges independent of σ whenever Re(ν) < −1.

2. L is a loop beginning and ending at +∞, encircling all poles of Γ(b_j − s), j = 1, 2, ..., m, exactly once in the negative direction, but not encircling any pole of Γ(1 − a_k + s), k = 1, 2, ..., n. Then the integral converges for all z if q > p ≥ 0; it also converges for q = p > 0 as long as |z| < 1. In the latter case, the integral additionally converges for |z| = 1 if Re(ν) < −1, where ν is defined as for the first path.

3. L is a loop beginning and ending at −∞ and encircling all poles of Γ(1 − a_k + s), k = 1, 2, ..., n, exactly once in the positive direction, but not encircling any pole of Γ(b_j − s), j = 1, 2, ..., m. Now the integral converges for all z if p > q ≥ 0; it also converges for p = q > 0 as long as |z| > 1. As noted for the second path too, in the case of p = q the integral also converges for |z| = 1 when Re(ν) < −1.

The conditions for convergence are readily established by applying Stirling's asymptotic approximation to the gamma functions in the integrand. When the integral converges for more than one of these paths, the results of integration can be shown to agree; if it converges for only one path, then this is the only one to be considered. In fact, numerical path integration in the complex plane constitutes a practicable and sensible approach to the calculation of Meijer G-functions.

As a consequence of this definition, the Meijer G-function is an analytic function of z with possible exception of the origin z = 0 and of the unit circle |z| = 1.