Meijer G-function

In mathematics, the G-function was introduced by 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 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 = G, 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 that can be written as a product G₁·G₂ 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.

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 :
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 for which k = 1, 2,..., n, and j = 1, 2,..., m, which implies that no pole of any Γ, j = 1, 2,..., m, coincides with any pole of any Γ, 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:
Implementations of the G-function in computer algebra systems typically employ separate vector arguments for the four 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:
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:
For a fundamental set of solutions of this equation in the case of p ≤ q one may take:
and similarly in the case of p ≥ q:
These particular solutions are analytic except for a possible singularity at z = 0, and in the case of p = q also an inevitable singularity at z = ^{p−''m−n''}. As will be seen presently, they can be identified with generalized hypergeometric functions _pF_q−1 of argument ^{p−''m−n''} z that are multiplied by a power z^b_h, and with generalized hypergeometric functions _qF_p−1 of argument ^{q−''m−n''} z⁻¹ that are multiplied by a power z^a_h−1, respectively.

Relationship between the G-function and the generalized hypergeometric function

If the integral converges when evaluated along the [|second path] introduced above, and if no confluent poles appear among the Γ, j = 1, 2,..., m, then the Meijer G-function can be expressed as a sum of residues in terms of generalized hypergeometric functions _pF_q−1 :
The star indicates that the term corresponding to j = h is omitted.
For the integral to converge along the second path one must have either p < q, or p = q and |z| < 1, and for the poles to be distinct no pair among the b_j, j = 1, 2,..., m, may differ by an integer or zero. The asterisks in the relation remind us to ignore the contribution with index j = h as follows: In the product this amounts to replacing Γ with 1, and in the argument of the hypergeometric function, if we recall the meaning of the vector notation,
this amounts to shortening the vector length from q to q−1.
Note that when m = 0, the second path does not contain any pole, and so the integral must vanish identically,
if either p < q, or p = q and |z| < 1.
Similarly, if the integral converges when evaluated along the [|third path] above, and if no confluent poles appear among the Γ, k = 1, 2,..., n, then the G-function can be expressed as:
For this, either p > q, or p = q and |z| > 1 are required, and no pair among the a_k, k = 1, 2,..., n, may differ by an integer or zero. For n = 0 one consequently has:
if either p > q, or p = q and |z| > 1.
On the other hand, any generalized hypergeometric function can readily be expressed in terms of the Meijer G-function:
where we have made use of the vector notation:
This holds unless a nonpositive integer value of at least one of its parameters a_p reduces the hypergeometric function to a finite polynomial, in which case the gamma prefactor of either G-function vanishes and the parameter sets of the G-functions violate the requirement a_k − b_j ≠ 1, 2, 3,... for k = 1, 2,..., n and j = 1, 2,..., m from the definition above. Apart from this restriction, the relationship is valid whenever the generalized hypergeometric series _pF_q converges, i. e. for any finite z when p ≤ q, and for |z| < 1 when p = q + 1. In the latter case, the relation with the G-function automatically provides the analytic continuation of _pF_q to |z| ≥ 1 with a branch cut from 1 to ∞ along the real axis. Finally, the relation furnishes a natural extension of the definition of the hypergeometric function to orders p > q + 1. By means of the G-function we can thus solve the generalized hypergeometric differential equation for p > q + 1 as well.

Polynomial cases

To express polynomial cases of generalized hypergeometric functions in terms of Meijer G-functions, a linear combination of two G-functions is needed in general:
where h = 0, 1, 2,... equals the degree of the polynomial _p+1F_q. The orders m and n can be chosen freely in the ranges 0 ≤ m ≤ q and 0 ≤ n ≤ p, which allows to avoid that specific integer values or integer differences among the parameters a_p and b_q of the polynomial give rise to divergent gamma functions in the prefactor or to a conflict with the [|definition of the G-function]. Note that the first G-function vanishes for n = 0 if p > q, while the second G-function vanishes for m = 0 if p < q. Again, the formula can be verified by expressing the two G-functions as sums of residues; no cases of confluent poles permitted by the definition of the G-function need be excluded here.

Basic properties of the G-function

As can be seen from the definition of the G-function, if equal parameters appear among the a_p and b_q determining the factors in the numerator and the denominator of the integrand, the fraction can be simplified, and the order of the function thereby be reduced. Whether the order m or n will decrease depends on the particular position of the parameters in question. Thus, if one of the a_k, k = 1, 2,..., n, equals one of the b_j, j = m + 1,..., q, the G-function lowers its orders p, q and n:
For the same reason, if one of the a_k, k = n + 1,..., p, equals one of the b_j, j = 1, 2,..., m, then the G-function lowers its orders p, q and m:
Starting from the definition, it is also possible to derive the following properties:
The abbreviations ν and δ were introduced in the definition of the G-function above.