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 zρ 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(cxγ), 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(cxγ)·G2(dxδ) 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):
G p , q m , n ( a 1 , … , a p b 1 , … , b q | z ) = 1 2 π i ∫ L ∏ j = 1 m Γ ( b j − s ) ∏ j = 1 n Γ ( 1 − a j + s ) ∏ j = m + 1 q Γ ( 1 − b j + s ) ∏ j = n + 1 p Γ ( a j − s ) z s d s , {\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:
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