In mathematics, the G-function was introduced by Cornelis Simon Meijer (1936) as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's G-function was able to include those as particular cases as well. The first definition was made by Meijer using a series; nowadays the accepted and more general definition is via a line integral in the complex plane, introduced in its full generality by Arthur Erdélyi in 1953.

With the modern definition, the majority of the established special functions can be represented in terms of the Meijer G-function. A notable property is the closure of the set of all G-functions not only under differentiation but also under indefinite integration. In combination with a functional equation that allows to liberate from a G-function G(z) any factor z^ρ that is a constant power of its argument z, the closure implies that whenever a function is expressible as a G-function of a constant multiple of some constant power of the function argument, f(x) = G(cx^γ), the derivative and the antiderivative of this function are expressible so too.

The wide coverage of special functions also lends power to uses of Meijer's G-function other than the representation and manipulation of derivatives and antiderivatives. For example, the definite integral over the positive real axis of any function g(x) that can be written as a product G₁(cx^γ)·G₂(dx^δ) of two G-functions with rational γ/δ equals just another G-function, and generalizations of integral transforms like the Hankel transform and the Laplace transform and their inverses result when suitable G-function pairs are employed as transform kernels.

A still more general function, which introduces additional parameters into Meijer's G-function, is Fox's H-function.

One application of the Meijer G-function has been the particle spectrum of radiation from an inertial horizon in the moving mirror model of the dynamical Casimir effect (Good 2020).

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):

${\displaystyle G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z\right)={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}-s)\prod _{j=1}^{n}\Gamma (1-a_{j}+s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}+s)\prod _{j=n+1}^{p}\Gamma (a_{j}-s)}}\,z^{s}\,ds,}$

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 any combination of {k, j} for which 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:

${\displaystyle G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z\right)=G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}\mathbf {a_{p}} \\\mathbf {b_{q}} \end{matrix}}\;\right|\,z\right).}$

Implementations 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

${\displaystyle \delta =m+n-{\tfrac {1}{2}}(p+q);}$

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

${\displaystyle \nu =\sum _{j=1}^{q}b_{j}-\sum _{j=1}^{p}a_{j}.}$

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.

Differential equation

The G-function satisfies the following linear differential equation of order max(p,q):

${\displaystyle \leftG(z)=0.}$

For a fundamental set of solutions of this equation in the case of p ≤ q one may take:

${\displaystyle G_{p,q}^{1,p}(\begin{array}{c}{a}_1,\dots <!--\; \dots \; -->,a_{}\end{array}}$Zdroj:https://en.wikipedia.org?pojem=Meijer_G-function
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