Meijer G-function

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.

With the modern definition, the majority of the established special functions can be represented in terms of the Meijer G-function.

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.

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).

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.

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 (possibly empty) parameter groups a1 ... an, an+1 ... ap, b1 ... bm, and bm+1 ... bq, and thus can omit the orders p, q, n, and m as redundant.

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.

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.

If the integral converges when evaluated along the second path introduced above, and if no confluent poles appear among the Γ(bj − s), j = 1, 2, ..., m, then the Meijer G-function can be expressed as a sum of residues in terms of generalized hypergeometric functions pFq−1 (Slater's theorem): The star indicates that the term corresponding to j = h is omitted.

In the latter case, the relation with the G-function automatically provides the analytic continuation of pFq(z) to |z| ≥ 1 with a branch cut from 1 to ∞ along the real axis.

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 ap and bq of the polynomial give rise to divergent gamma functions in the prefactor or to a conflict with the definition of the G-function.

As can be seen from the definition of the G-function, if equal parameters appear among the ap and bq 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.

As usual, the inverse transform is then given by: where c is a real positive constant that places the integration path to the right of any pole in the integrand.

Roop Narain (1962, 1963a, 1963b) showed that the functions: are an asymmetric pair of transform kernels, where γ > 0, n − p = m − q > 0, and: along with further convergence conditions.