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{{Short description|Generalization of the hypergeometric function}}
[[File:Plot of the Meiger G function G(((a 1,...,a n),(a n+1,...,a p)),((b 1,...,b m),(b m+1,...,b q)),z) input ((½),()),((⅓),()) in the complex plane from -2-2i to 2+2i.svg|alt=Plot of the Meijer G function G(((a 1,...,a n),(a n+1,...,a p)),((b 1,...,b m),(b m+1,...,b q)),z) input (({{sfrac|1|2}}),()),(({{sfrac|1|3}}),()) in the complex plane from -2-2i to 2+2i|thumb|Plot of the Meijer G function G(((a 1,...,a n),(a n+1,...,a p)),((b 1,...,b m),(b m+1,...,b q)),z) input (({{sfrac|1|2}}),()),(({{sfrac|1|3}}),()) in the complex plane from -2-2i to 2+2i]]
In mathematics, the '''G-function''' was introduced by {{harvs | txt | authorlink= Cornelis Simon Meijer | first= Cornelis Simon | last= Meijer | year= 1936}} as a very general [[function (mathematics)|function]] intended to include most of the known [[special function]]s 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 (mathematics)|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 (mathematics)|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''<sup>''ρ''</sup> 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''<sup>''γ''</sup>), 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 ''G''<sub>1</sub>(''cx''<sup>''γ''</sup>)·''G''<sub>2</sub>(''dx''<sup>''δ''</sup>) of two G-functions with [[rational number|rational]] ''γ''/''δ'' equals just another G-function, and generalizations of [[integral transform]]s 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 H-function|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 {{harv|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]] {{harv|Bateman|Erdélyi|1953|loc=§ 5.3-1}}:

:<math>
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,
</math>

where Γ denotes the [[gamma function]]. This integral is of the so-called [[Mellin–Barnes integral|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''<sub>''k''</sub> − ''b''<sub>''j''</sub> ≠ 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 (complex analysis)|pole]] of any Γ(''b''<sub>''j''</sub> − ''s''), ''j'' = 1, 2, ..., ''m'', coincides with any pole of any Γ(1 − ''a''<sub>''k''</sub> + ''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 [[vector (mathematics and physics)|vectors]]:

:<math>
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) .
</math>

Implementations of the G-function in [[computer algebra system]]s typically employ separate vector arguments for the four (possibly empty) parameter groups ''a''<sub>1</sub> ... ''a''<sub>''n''</sub>, ''a''<sub>''n''+1</sub> ... ''a''<sub>''p''</sub>, ''b''<sub>1</sub> ... ''b''<sub>''m''</sub>, and ''b''<sub>''m''+1</sub> ... ''b''<sub>''q''</sub>, 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''<sub>''j''</sub> − ''s''), ''j'' = 1, 2, ..., ''m'', are on the right of the path, while all poles of Γ(1 − ''a''<sub>''k''</sub> + ''s''), ''k'' = 1, 2, ..., ''n'', are on the left. The integral then converges for |arg ''z''| < ''δ'' ''π'', where
::<math>
\delta = m + n - \tfrac{1}{2} (p+q) ;
</math>
:an obvious prerequisite for this is ''δ'' > 0. The integral additionally converges for |arg ''z''| = ''δ'' ''π'' ≥ 0 if (q − p) (''σ'' + <sup>1</sup>⁄<sub>2</sub>) > Re(''ν'') + 1, where ''σ'' represents Re(''s'') as the integration variable ''s'' approaches both +''i''∞ and −''i''∞, and where
::<math>
\nu = \sum_{j = 1}^q b_j - \sum_{j = 1}^p a_j .
</math>
: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''<sub>''j''</sub> − ''s''), ''j'' = 1, 2, ..., ''m'', exactly once in the negative direction, but not encircling any pole of Γ(1 − ''a''<sub>''k''</sub> + ''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''<sub>''k''</sub> + ''s''), ''k'' = 1, 2, ..., ''n'', exactly once in the positive direction, but not encircling any pole of Γ(''b''<sub>''j''</sub> − ''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 approximation|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 [[ordinary differential equation|differential equation]] of order max(''p'',''q''):

:<math>
\left[ (-1)^{p - m - n} \;z \prod_{j = 1}^p \left( z \frac{d}{dz} - a_j + 1 \right) - \prod_{j = 1}^q \left( z \frac{d}{dz} - b_j \right) \right] G(z) = 0.
</math>

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

:<math>
G_{p,q}^{\,1,p} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\ b_h, b_1, \dots, b_{h-1}, b_{h+1}, \dots, b_q \end{matrix} \; \right| \, (-1)^{p-m-n+1} \;z \right), \quad h = 1,2,\dots,q,
</math>

and similarly in the case of ''p'' ≥ ''q'':

:<math>
G_{p,q}^{\,q,1} \!\left( \left. \begin{matrix} a_h, a_1, \dots, a_{h-1}, a_{h+1}, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \; \right| \, (-1)^{q-m-n+1} \;z \right), \quad h = 1,2,\dots,p.
</math>

These particular solutions are analytic except for a possible [[mathematical singularity|singularity]] at ''z'' = 0 (as well as a possible singularity at ''z'' = ∞), and in the case of ''p'' = ''q'' also an inevitable singularity at ''z'' = (−1)<sup>''p''−''m''−''n''</sup>. As will be seen presently, they can be identified with [[generalized hypergeometric function]]s <sub>''p''</sub>''F''<sub>''q''−1</sub> of argument (−1)<sup>''p''−''m''−''n''</sup> ''z'' that are multiplied by a power ''z''<sup>''b''<sub>''h''</sub></sup>, and with generalized hypergeometric functions <sub>''q''</sub>''F''<sub>''p''−1</sub> of argument (−1)<sup>''q''−''m''−''n''</sup> ''z''<sup>&minus;1</sup> that are multiplied by a power ''z''<sup>''a''<sub>''h''</sub>−1</sup>, respectively.

==Relationship between the G-function and the generalized hypergeometric function==

If the integral converges when evaluated along the [[#Definition of the Meijer G-function|second path]] introduced above, and if no confluent [[pole (complex analysis)|poles]] appear among the Γ(''b''<sub>''j''</sub> − ''s''), ''j'' = 1, 2, ..., ''m'', then the Meijer G-function can be expressed as a sum of [[residue (complex analysis)|residue]]s in terms of [[generalized hypergeometric function]]s <sub>''p''</sub>''F''<sub>''q''−1</sub> (Slater's theorem):

:<math>
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = \sum_{h=1}^m \frac{\prod_{j=1}^m \Gamma(b_j - b_h)^* \prod_{j=1}^n \Gamma(1+b_h - a_j) \; z^{b_h}} {\prod_{j=m+1}^q \Gamma(1+b_h - b_j) \prod_{j=n+1}^p \Gamma(a_j - b_h)} \times
</math>
:<math>
_{p}F_{q-1} \!\left( \left. \begin{matrix} 1+b_h - \mathbf{a_p} \\ (1+b_h - \mathbf{b_q})^* \end{matrix} \; \right| \, (-1)^{p-m-n} \; z \right) .
</math>
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''<sub>''j''</sub>, ''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(0) with 1, and in the argument of the hypergeometric function, if we recall the meaning of the vector notation,

:<math>
1 + b_h - \mathbf{b_q} = (1 + b_h - b_1), \,\dots, \,(1 + b_h - b_j), \,\dots, \,(1 + b_h - b_q),
</math>

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,

:<math>
G_{p,q}^{\,0,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = 0,
</math>

if either ''p'' < ''q'', or ''p'' = ''q'' and |''z''| < 1.

Similarly, if the integral converges when evaluated along the [[#Definition of the Meijer G-function|third path]] above, and if no confluent poles appear among the Γ(1 − ''a''<sub>''k''</sub> + ''s''), ''k'' = 1, 2, ..., ''n'', then the G-function can be expressed as:

:<math>
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = \sum_{h=1}^n \frac{\prod_{j=1}^n \Gamma(a_h - a_j)^* \prod_{j=1}^m \Gamma(1-a_h + b_j) \; z^{a_h-1}} {\prod_{j=n+1}^p \Gamma(1-a_h + a_j) \prod_{j=m+1}^q \Gamma(a_h - b_j)} \times
</math>
:<math>
 _{q}F_{p-1} \!\left( \left. \begin{matrix} 1-a_h + \mathbf{b_q} \\ (1-a_h + \mathbf{a_p})^* \end{matrix} \; \right| \, (-1)^{q-m-n} z^{-1} \right) .
</math>

For this, either ''p'' > ''q'', or ''p'' = ''q'' and |''z''| > 1 are required, and no pair among the ''a''<sub>''k''</sub>, ''k'' = 1, 2, ..., ''n'', may differ by an integer or zero. For ''n'' = 0 one consequently has:

:<math>
G_{p,q}^{\,m,0} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = 0,
</math>

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:

:<math>
\; _{p}F_{q} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right)
= \frac {\Gamma(\mathbf{b_q})} {\Gamma(\mathbf{a_p})} \; G_{p,\,q+1}^{\,1,\,p} \!\left( \left. \begin{matrix} 1-\mathbf{a_p} \\ 0,1 - \mathbf{b_q} \end{matrix} \; \right| \, -z \right)
= \frac {\Gamma(\mathbf{b_q})} {\Gamma(\mathbf{a_p})} \; G_{q+1,\,p}^{\,p,\,1} \!\left( \left. \begin{matrix} 1,\mathbf{b_q} \\ \mathbf{a_p} \end{matrix} \; \right| \, -z^{-1} \right),
</math>

where we have made use of the vector notation:

:<math>
\Gamma(\mathbf{a_p}) = \prod_{j=1}^p \Gamma(a_j).
</math>

This holds unless a nonpositive integer value of at least one of its parameters '''a'''<sub>'''p'''</sub> 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''<sub>''k''</sub> − ''b''<sub>''j''</sub> ≠ 1, 2, 3, ... for ''k'' = 1, 2, ..., ''n'' and ''j'' = 1, 2, ..., ''m'' from the [[#Definition of the Meijer G-function|definition]] above. Apart from this restriction, the relationship is valid whenever the generalized hypergeometric series <sub>''p''</sub>''F''<sub>''q''</sub>(''z'') 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 <sub>''p''</sub>''F''<sub>''q''</sub>(''z'') 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:

:<math>
\; _{p+1}F_{q} \!\left( \left. \begin{matrix} -h, \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = h! \; \frac{\prod_{j=n+1}^p \Gamma(1 - a_j) \prod_{j=m+1}^q \Gamma(b_j)} {\prod_{j=1}^n \Gamma(a_j) \prod_{j=1}^m \Gamma(1 - b_j)} \times
</math>
:<math>
\left[ G_{p+1,\,q+1}^{\,m+1,\,n} \!\left( \left. \begin{matrix} 1-\mathbf{a_p}, h+1 \\ 0, 1-\mathbf{b_q} \end{matrix} \; \right| \, (-1)^{p-m-n} \; z \right) + (-1)^h \; G_{p+1,\,q+1}^{\,m,\,n+1} \!\left( \left. \begin{matrix} h+1, 1-\mathbf{a_p} \\ 1-\mathbf{b_q}, 0 \end{matrix} \; \right| \, (-1)^{p-m-n} \; z \right) \right] ,
</math>

where ''h'' = 0, 1, 2, ... equals the degree of the polynomial <sub>''p''+1</sub>''F''<sub>''q''</sub>(''z''). 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'''<sub>'''p'''</sub> and '''b'''<sub>'''q'''</sub> of the polynomial give rise to divergent gamma functions in the prefactor or to a conflict with the [[#Definition of the Meijer G-function|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 [[residue (complex analysis)|residues]]; no cases of confluent [[pole (complex analysis)|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 Meijer G-function|definition of the G-function]], if equal parameters appear among the '''a'''<sub>'''p'''</sub> and '''b'''<sub>'''q'''</sub> 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''<sub>''k''</sub>, ''k'' = 1, 2, ..., ''n'', equals one of the ''b''<sub>''j''</sub>, ''j'' = ''m'' + 1, ..., ''q'', the G-function lowers its orders ''p'', ''q'' and ''n'':

:<math>
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, a_2, \dots, a_p \\ b_1, \dots, b_{q-1}, a_1 \end{matrix} \; \right| \, z \right) =
G_{p-1,\,q-1}^{\,m,\,n-1} \!\left( \left. \begin{matrix} a_2, \dots, a_p \\ b_1, \dots, b_{q-1} \end{matrix} \; \right| \, z \right), \quad n,p,q \geq 1.
</math>

For the same reason, if one of the ''a''<sub>''k''</sub>, ''k'' = ''n'' + 1, ..., ''p'', equals one of the ''b''<sub>''j''</sub>, ''j'' = 1, 2, ..., ''m'', then the G-function lowers its orders ''p'', ''q'' and ''m'':

:<math>
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_{p-1}, b_1 \\ b_1, b_2, \dots, b_q \end{matrix} \; \right| \, z \right) =
G_{p-1,\,q-1}^{\,m-1,\,n} \!\left( \left. \begin{matrix} a_1, \dots, a_{p-1} \\ b_2, \dots, b_q \end{matrix} \; \right| \, z \right), \quad m,p,q \geq 1.
</math>

Starting from the definition, it is also possible to derive the following properties:

:<math>
z^{\rho} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) =
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} + \rho \\ \mathbf{b_q} + \rho \end{matrix} \; \right| \, z \right),
</math>

:<math>
G_{p+2,\,q}^{\,m,\,n+1} \!\left( \left. \begin{matrix} \alpha, \mathbf{a_p}, \alpha' \\ \mathbf{b_q} \end{matrix} \; \right| \, z  \right) =
(-1)^{\alpha'-\alpha} \; G_{p+2,\,q}^{\,m,\,n+1} \!\left( \left. \begin{matrix} \alpha', \mathbf{a_p}, \alpha \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right), \quad n \leq p, \; \alpha'-\alpha \in \mathbb{Z},
</math>

:<math>
G_{p,\,q+2}^{\,m+1,\,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \beta, \mathbf{b_q}, \beta' \end{matrix} \; \right| \, z \right) =
(-1)^{\beta'-\beta} \; G_{p,\,q+2}^{\,m+1,\,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \beta', \mathbf{b_q}, \beta \end{matrix} \; \right| \, z \right), \quad m \leq q, \; \beta'-\beta \in \mathbb{Z},
</math>

:<math>
G_{p+1,\,q+1}^{\,m,\,n+1} \!\left( \left. \begin{matrix} \alpha, \mathbf{a_p} \\ \mathbf{b_q}, \beta \end{matrix} \; \right| \, z \right) =
(-1)^{\beta-\alpha} \; G_{p+1,\,q+1}^{\,m+1,\,n} \!\left( \left. \begin{matrix} \mathbf{a_p}, \alpha \\ \beta, \mathbf{b_q} \end{matrix} \; \right| \, z \right), \quad m \leq q, \; \beta-\alpha = 0,1,2,\dots,
</math>

:<math>
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) =
G_{q,p}^{\,n,m} \!\left( \left. \begin{matrix} 1-\mathbf{b_q} \\ 1-\mathbf{a_p} \end{matrix} \; \right| \, z^{-1} \right),
</math>

:<math>
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) =
\frac{h^{1+\nu+(p-q)/2}} {(2 \pi)^{(h-1) \delta}} \; G_{h p, \, h q}^{\, h m, \, h n} \!\left( \left. \begin{matrix} a_1/h, \dots, (a_1+h-1)/h, \dots, a_p/h, \dots, (a_p+h-1)/h \\ b_1/h, \dots, (b_1+h-1)/h, \dots, b_q/h, \dots, (b_q+h-1)/h \end{matrix} \; \right| \, \frac{z^h} {h^{h(q-p)}} \right), \quad h \in \mathbb{N}.
</math>

The abbreviations ''ν'' and ''δ'' were introduced in the [[#Definition of the Meijer G-function|definition of the G-function]] above.

===Derivatives and antiderivatives===

Concerning [[derivative]]s of the G-function, one finds these relationships:

:<math>
\frac{d}{dz} \left[ z^{1-a_1} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) \right] =
z^{-a_1} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1 - 1, a_2, \dots, a_p \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right), \quad n \geq 1,
</math>

:<math>
\frac{d}{dz} \left[ z^{1-a_p} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) \right] =
- z^{-a_p} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_{p-1}, a_p - 1 \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right), \quad n < p.
</math>

:<math>
\frac{d}{dz} \left[ z^{-b_1} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) \right] =
- z^{-1-b_1} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ b_1 + 1, b_2, \dots, b_q \end{matrix} \; \right| \, z \right), \quad m \geq 1,
</math>

:<math>
\frac{d}{dz} \left[ z^{-b_q} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) \right] =
z^{-1-b_q} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ b_1, \dots, b_{q-1}, b_q + 1 \end{matrix} \; \right| \, z \right), \quad m < q,
</math>

From these four, equivalent relations can be deduced by simply evaluating the derivative on the left-hand side and manipulating a bit. One obtains for example:

:<math>
z \frac{d}{dz} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) =
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1 -1, a_2, \dots, a_p \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) +
(a_1 - 1) \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right), \quad n \geq 1.
</math>

Moreover, for derivatives of arbitrary order ''h'', one has

:<math>
z^h \frac{d^h}{dz^h} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) =
G_{p+1,\,q+1}^{\,m,\,n+1} \!\left( \left. \begin{matrix} 0, \mathbf{a_p} \\ \mathbf{b_q}, h \end{matrix} \; \right| \, z \right) =
(-1)^h \; G_{p+1,\,q+1}^{\,m+1,\,n} \!\left( \left. \begin{matrix} \mathbf{a_p}, 0 \\ h, \mathbf{b_q} \end{matrix} \; \right| \, z \right),
</math>

:<math>
z^h \frac{d^h}{dz^h} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z^{-1} \right) =
G_{p+1,\,q+1}^{\,m+1,\,n} \!\left( \left. \begin{matrix} \mathbf{a_p}, 1-h \\ 1, \mathbf{b_q} \end{matrix} \; \right| \, z^{-1} \right) =
(-1)^h \; G_{p+1,\,q+1}^{\,m,\,n+1} \!\left( \left. \begin{matrix} 1-h, \mathbf{a_p} \\ \mathbf{b_q}, 1 \end{matrix} \; \right| \, z^{-1} \right),
</math>

which hold for ''h'' < 0 as well, thus allowing to obtain the [[antiderivative]] of any G-function as easily as the derivative. By choosing one or the other of the two results provided in either formula, one can always prevent the set of parameters in the result from violating the condition ''a''<sub>''k''</sub> − ''b''<sub>''j''</sub> ≠ 1, 2, 3, ... for ''k'' = 1, 2, ..., ''n'' and ''j'' = 1, 2, ..., ''m'' that is imposed by the [[#Definition of the Meijer G-function|definition of the G-function]]. Note that each pair of results becomes unequal in the case of ''h'' < 0.

From these relationships, corresponding properties of the [[hypergeometric function|Gauss hypergeometric function]] and of other special functions can be derived.

===Recurrence relations===

By equating different expressions for the first-order derivatives, one arrives at the following 3-term recurrence relations among contiguous G-functions:

:<math>
(a_p - a_1) \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) =
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1-1, a_2, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right) +
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_{p-1}, a_p-1 \\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right), \quad 1 \leq n < p,
</math>

:<math>
(b_1 - b_q) \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) =
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\ b_1+1, b_2, \dots, b_q \end{matrix} \; \right| \, z \right) +
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_{q-1}, b_q+1 \end{matrix} \; \right| \, z \right), \quad 1 \leq m < q,
</math>

:<math>
(b_1 - a_1 + 1) \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) =
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1-1, a_2, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right) +
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\ b_1+1, b_2, \dots, b_q \end{matrix} \; \right| \, z \right), \quad n \geq 1, \; m \geq 1,
</math>

:<math>
(a_p - b_q - 1) \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) =
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_{p-1}, a_p-1 \\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right) +
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_{q-1}, b_q+1 \end{matrix} \; \right| \, z \right), \quad n < p, \; m < q.
</math>

Similar relations for the diagonal parameter pairs ''a''<sub>1</sub>, ''b''<sub>''q''</sub> and ''b''<sub>1</sub>, ''a''<sub>''p''</sub> follow by suitable combination of the above. Again, corresponding properties of hypergeometric and other special functions can be derived from these recurrence relations.

===Multiplication theorems===

Provided that ''z'' ≠ 0, the following relationships hold:

:<math>
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, w z \right) =
w^{b_1} \sum_{h=0}^{\infty} \frac{(1 - w)^h}{h!} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ b_1+h, b_2, \dots, b_q \end{matrix} \; \right| \, z \right), \quad m \geq 1,
</math>

:<math>
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, w z \right) =
w^{b_q} \sum_{h=0}^{\infty} \frac{(w - 1)^h}{h!} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ b_1, \dots, b_{q-1}, b_q+h \end{matrix} \; \right| \, z \right), \quad m < q,
</math>

:<math>
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, \frac{z}{w} \right) =
w^{1-a_1} \sum_{h=0}^{\infty} \frac{(1 - w)^h}{h!} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1-h, a_2, \dots, a_p \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right), \quad n \geq 1,
</math>

:<math>
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, \frac{z}{w} \right) =
w^{1-a_p} \sum_{h=0}^{\infty} \frac{(w - 1)^h}{h!} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_{p-1}, a_p-h \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right), \quad n < p.
</math>

These follow by [[Taylor series|Taylor expansion]] about ''w'' = 1, with the help of the [[#Basic properties of the G-function|basic properties]] discussed above. The [[radius of convergence|radii of convergence]] will be dependent on the value of ''z'' and on the G-function that is expanded. The expansions can be regarded as generalizations of similar theorems for [[Bessel function|Bessel]], [[hypergeometric function|hypergeometric]] and [[confluent hypergeometric function|confluent hypergeometric]] functions.

==Definite integrals involving the G-function==

Among [[integral|definite integrals]] involving an arbitrary G-function one has:

:<math>
\int_0^{\infty} x^{s - 1} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, \eta x \right) dx =
\frac{\eta^{-s} \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)}.
</math>

Note that the restrictions under which this integral exists have been omitted here. It is, of course, no surprise that the [[Mellin transform]] of a G-function should lead back to the integrand appearing in the [[#Definition of the Meijer G-function|definition]] above.

[[Leonhard Euler|Euler]]-type integrals for the G-function are given by:

:<math>
\int_0^1 x^{-\alpha} \; (1-x)^{\alpha - \beta - 1} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z x \right) dx =
\Gamma (\alpha - \beta) \; G_{p+1 ,\, q+1}^{\,m ,\, n+1} \!\left( \left. \begin{matrix} \alpha, \mathbf{a_p} \\ \mathbf{b_q}, \beta \end{matrix} \; \right| \, z \right),
</math>

:<math>
\int_1^\infty x^{-\alpha} \; (x-1)^{\alpha - \beta - 1} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z x \right) dx =
\Gamma (\alpha - \beta) \; G_{p+1 ,\, q+1}^{\,m+1 ,\, n} \!\left( \left. \begin{matrix} \mathbf{a_p}, \alpha \\ \beta, \mathbf{b_q} \end{matrix} \; \right| \, z \right).
</math>

Extensive restrictions under which these integrals exist can be found on p.&nbsp;417 of "Tables of Integral Transforms", vol. II(1954), Edited by A. Erdelyi. Note that, in view of their effect on the G-function, these integrals can be used to define the operation of [[fractional calculus|fractional integration]] for a fairly large class of functions ([[Erdélyi–Kober operator]]s).

A result of fundamental importance is that the product of two arbitrary G-functions integrated over the positive real axis can be represented by just another G-function (convolution theorem):

:<math>
\int_0^{\infty} G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, \eta x \right)
G_{\sigma, \tau}^{\,\mu, \nu} \!\left( \left. \begin{matrix} \mathbf{c_{\sigma}} \\ \mathbf{d_\tau} \end{matrix} \; \right| \, \omega x \right) dx =
</math>
:<math>
= \frac{1}{\eta} \; G_{q + \sigma ,\, p + \tau}^{\,n + \mu ,\, m + \nu} \!\left( \left. \begin{matrix} - b_1, \dots, - b_m, \mathbf{c_{\sigma}}, - b_{m+1}, \dots, - b_q \\ - a_1, \dots, -a_n, \mathbf{d_\tau} , - a_{n+1}, \dots, - a_p \end{matrix} \; \right| \, \frac{\omega}{\eta} \right) =
</math>
:<math> = \frac{1}{\omega} \; G_{p + \tau ,\, q + \sigma}^{\,m + \nu ,\, n + \mu} \!\left( \left. \begin{matrix} a_1, \dots, a_n, -\mathbf{d_\tau} , a_{n+1}, \dots, a_p \\ b_1, \dots, b_m, -\mathbf{c_{\sigma}}, b_{m+1}, \dots, b_q \end{matrix} \; \right| \, \frac{\eta}{\omega} \right) .
</math>

Restrictions under which the integral exists can be found in Meijer, C. S., 1941: Nederl. Akad. Wetensch, Proc. 44, pp.&nbsp;82–92. Note how the Mellin transform of the result merely assembles the gamma factors from the Mellin transforms of the two functions in the integrand.

The convolution formula can be derived by substituting the defining Mellin–Barnes integral for one of the G-functions, reversing the order of integration, and evaluating the inner Mellin-transform integral. The preceding Euler-type integrals follow analogously.

===Laplace transform===

Using the above [[#Definite integrals involving the G-function|convolution integral]] and [[#Basic properties of the G-function|basic properties]] one can show that:

:<math>
\int_0^{\infty} e^{- \omega x} \; x^{- \alpha} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, \eta x \right) dx =
\omega^{\alpha - 1} \; G_{p + 1,\,q}^{\,m,\,n+1} \!\left( \left. \begin{matrix} \alpha, \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, \frac{\eta}{\omega} \right) ,
</math>

where Re(''ω'') > 0. This is the [[Laplace transform]] of a function ''G''(''ηx'') multiplied by a power ''x''<sup>−''α''</sup>; if we put ''α'' = 0 we get the Laplace transform of the G-function. As usual, the inverse transform is then given by:

:<math>
x^{- \alpha} \; G_{p,\,q+1}^{\,m,\,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q}, \alpha \end{matrix} \; \right| \, \eta x \right) = 
\frac{1}{2 \pi i} \int_{c - i \infty}^{c + i \infty} e^{\omega x} \; \omega^{\alpha - 1} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, \frac{\eta}{\omega} \right) d\omega,
</math>

where ''c'' is a real positive constant that places the integration path to the right of any [[pole (complex analysis)|pole]] in the integrand.

Another formula for the Laplace transform of a G-function is:

:<math>
\int_{0}^{\infty} e^{- \omega x} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, \eta x^2 \right) dx =
\frac{1}{\sqrt{\pi} \omega} \; G_{p+2,\,q}^{\,m,\,n+2} \!\left( \left. \begin{matrix} 0, \frac{1}{2}, \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, \frac{4 \eta}{\omega^2} \right) ,
</math>

where again Re(''ω'') > 0. Details of the restrictions under which the integrals exist have been omitted in both cases.

==Integral transforms based on the G-function==

In general, two functions ''k''(''z'',''y'') and ''h''(''z'',''y'') are called a pair of transform kernels if, for any suitable function ''f''(''z'') or any suitable function ''g''(''z''), the following two relationships hold simultaneously:

:<math>
g(z) = \int_{0}^{\infty} k(z,y) \, f(y) \; dy, \quad
f(z) = \int_{0}^{\infty} h(z,y) \, g(y) \; dy.
</math>

The pair of kernels is said to be symmetric if ''k''(''z'',''y'') = ''h''(''z'',''y'').

===Narain transform===

{{harvs | txt | first= Roop | last= Narain | year= 1962 | year2= 1963a | year3= 1963b}} showed that the functions:

:<math>
k(z,y) = 2 \gamma \; (zy)^{\gamma - 1/2} \; G_{p+q,\,m+n}^{\,m,\,p} \!\left( \left. \begin{matrix} \mathbf{a_p}, \mathbf{b_q} \\ \mathbf{c_m}, \mathbf{d_n} \end{matrix} \; \right| \, (zy)^{2 \gamma} \right),
</math>

:<math>
h(z,y) = 2 \gamma \; (zy)^{\gamma - 1/2} \; G_{p+q,\,m+n}^{\,n,\,q} \!\left( \left. \begin{matrix} -\mathbf{b_q}, -\mathbf{a_p} \\ -\mathbf{d_n}, -\mathbf{c_m} \end{matrix} \; \right| \, (zy)^{2 \gamma} \right)
</math>

are an asymmetric pair of transform kernels, where ''γ'' > 0, ''n'' − ''p'' = ''m'' − ''q'' > 0, and:

:<math>
\sum_{j=1}^p a_j + \sum_{j=1}^q b_j = \sum_{j=1}^m c_j + \sum_{j=1}^n d_j,
</math>

along with further convergence conditions. In particular, if ''p'' = ''q'', ''m'' = ''n'', ''a''<sub>''j''</sub> + ''b''<sub>''j''</sub> = 0 for ''j'' = 1, 2, ..., ''p'' and ''c''<sub>''j''</sub> + ''d''<sub>''j''</sub> = 0 for ''j'' = 1, 2, ..., ''m'', then the pair of kernels becomes symmetric. The well-known [[Hankel transform]] is a symmetric special case of the Narain transform (''γ'' = 1, ''p'' = ''q'' = 0, ''m'' = ''n'' = 1, ''c''<sub>1</sub> = −''d''<sub>1</sub> = <sup>''ν''</sup>⁄<sub>2</sub>).

===Wimp transform===
{{harvs | txt | first= Jet | last= Wimp | year= 1964}} showed that these functions are an asymmetric pair of transform kernels:

:<math>
k(z,y) = G_{p+2,\,q}^{\,m,\,n+2} \!\left( \left. \begin{matrix} 1 - \nu + i z, 1 - \nu - i z, \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; y \right),
</math>

:<math>
h(z,y) = \frac{i}{\pi} y e^{- \nu \pi i} \left[ e^{\pi y} A(\nu + i y, \nu - i y \,|\, z e^{i \pi} ) - e^{- \pi y} A(\nu - i y, \nu + i y \,|\, z e^{i \pi} ) \right],
</math>

where the function ''A''(·) is defined as:

:<math>
A(\alpha, \beta \,|\, z) = G_{p+2,\,q}^{\,q-m,\,p-n+1} \!\left( \left. \begin{matrix} -a_{n+1}, -a_{n+2}, \dots, -a_p, \alpha, -a_1, -a_2, \dots, -a_n, \beta \\ -b_{m+1}, -b_{m+2}, \dots, -b_q, -b_1, -b_2, \dots, -b_m \end{matrix} \; \right| \, z \right).
</math>

===Generalized Laplace transform===

The [[Laplace transform]] can be generalized in close analogy with Narain's generalization of the Hankel transform:

:<math>
g(s) = 2 \gamma \int_0^{\infty} (st)^{\gamma + \rho - 1/2} \; G_{p,\,q+1}^{\,q+1,\,0} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ 0, \mathbf{b_q} \end{matrix} \; \right| \, (st)^{2 \gamma} \right) f(t) \; dt,
</math>

:<math>
f(t) = \frac {\gamma} {\pi i} \int_{c - i \infty}^{c + i \infty} (ts)^{\gamma - \rho - 1/2} \; G_{p,\,q+1}^{\,1,\,p} \!\left( \left. \begin{matrix} -\mathbf{a_p} \\ 0, -\mathbf{b_q} \end{matrix} \; \right| \, -(ts)^{2 \gamma} \right) g(s) \; ds,
</math>

where ''γ'' > ''0'', ''p'' ≤ ''q'', and:

:<math>
(q+1-p) \, {\rho \over 2 \gamma} = \sum_{j=1}^p a_j - \sum_{j=1}^q b_j,
</math>

and where the constant ''c'' > 0 places the second integration path to the right of any pole in the integrand. For ''γ'' = <sup>1</sup>⁄<sub>2</sub>, ''ρ'' = 0 and ''p'' = ''q'' = 0, this corresponds to the familiar Laplace transform.

===Meijer transform===

Two particular cases of this generalization were given by C.S. Meijer in 1940 and 1941. The case resulting for ''γ'' = 1, ''ρ'' = −''ν'', ''p'' = 0, ''q'' = 1 and ''b''<sub>1</sub> = ''ν'' may be written  {{harvs | last= Meijer | year= 1940}}:

:<math>
g(s) = \sqrt {2 / \pi} \int_0^{\infty} (st)^{1/2} \, K_{\nu}(st) \, f(t) \; dt,
</math>

:<math>
f(t) = \frac {1} {\sqrt {2 \pi} \,i} \int_{c - i \infty}^{c + i \infty} (ts)^{1/2} \, I_{\nu}(ts) \, g(s) \; ds,
</math>

and the case obtained for ''γ'' = <sup>1</sup>⁄<sub>2</sub>, ''ρ'' = −''m'' − ''k'', ''p'' = ''q'' = 1, ''a''<sub>1</sub> = ''m'' − ''k'' and ''b''<sub>1</sub> = 2''m'' may be written {{harvs | last= Meijer | year= 1941a}}:

:<math>
g(s) = \int_0^{\infty} (st)^{-k-1/2} \, e^{-st/2} \, W_{k+1/2,\,m}(st) \, f(t) \; dt,
</math>

:<math>
f(t) = \frac {\Gamma(1-k+m)} {2 \pi i \, \Gamma(1+2m)} \int_{c - i \infty}^{c + i \infty} (ts)^{k-1/2} \, e^{ts/2} \, M_{k-1/2,\,m}(ts) \, g(s) \; ds.
</math>

Here ''I''<sub>''ν''</sub> and ''K''<sub>''ν''</sub> are the [[Bessel function|modified Bessel functions]] of the first and second kind, respectively, ''M''<sub>''k'',''m''</sub> and ''W''<sub>''k'',''m''</sub> are the [[Whittaker function]]s, and constant scale factors have been applied to the functions ''f'' and ''g'' and their arguments ''s'' and ''t'' in the first case.

==Representation of other functions in terms of the G-function==

The following list shows how the familiar [[elementary function]]s result as special cases of the Meijer G-function:

:<math> e^x = G_{0,1}^{\,1,0} \!\left( \left. \begin{matrix} - \\ 0 \end{matrix} \; \right| \, -x \right), \qquad \forall x </math>

:<math> \cos x = \sqrt{\pi} \; G_{0,2}^{\,1,0} \!\left( \left. \begin{matrix} - \\ 0,\frac{1}{2} \end{matrix} \; \right| \, \frac{x^2}{4} \right), \qquad \forall x </math>

:<math> \sin x = \sqrt{\pi} \; G_{0,2}^{\,1,0} \!\left( \left. \begin{matrix} - \\ \frac{1}{2},0 \end{matrix} \; \right| \, \frac{x^2}{4} \right), \qquad \frac{-\pi}{2} < \arg x \leq \frac{\pi}{2} </math>

:<math> \cosh x = \sqrt{\pi} \; G_{0,2}^{\,1,0} \!\left( \left. \begin{matrix} - \\ 0,\frac{1}{2} \end{matrix} \; \right| \, -\frac{x^2}{4} \right), \qquad \forall x </math>

:<math> \sinh x = -\sqrt{\pi}i \; G_{0,2}^{\,1,0} \!\left( \left. \begin{matrix} - \\ \frac{1}{2},0 \end{matrix} \; \right| \, -\frac{x^2}{4} \right), \qquad -\pi < \arg x \leq 0 </math>

:<math> \arcsin x = \frac{-i}{2\sqrt{\pi}} \; G_{2,2}^{\,1,2} \!\left( \left. \begin{matrix} 1,1 \\ \frac{1}{2},0 \end{matrix} \; \right| \, -x^2 \right), \qquad -\pi < \arg x \leq 0 </math>

:<math> \arctan x = \frac{1}{2} \; G_{2,2}^{\,1,2} \!\left( \left. \begin{matrix} \frac{1}{2},1 \\ \frac{1}{2},0 \end{matrix} \; \right| \, x^2 \right), \qquad \frac{-\pi}{2} < \arg x \leq \frac{\pi}{2} </math>

:<math> \arccot x = \frac{1}{2} \; G_{2,2}^{\,2,1} \!\left( \left. \begin{matrix} \frac{1}{2},1 \\ \frac{1}{2},0 \end{matrix} \; \right| \, x^2 \right), \qquad \frac{-\pi}{2} < \arg x \leq \frac{\pi}{2} </math>

:<math> \ln (1+x) = G_{2,2}^{\,1,2} \!\left( \left. \begin{matrix} 1,1 \\ 1,0 \end{matrix} \; \right| \, x \right), \qquad \forall x </math>

:<math> H(1-|x|) = G_{1,1}^{\,1,0} \!\left( \left. \begin{matrix} 1 \\ 0 \end{matrix} \; \right| \, x \right), \qquad \forall x </math>

:<math> H(|x|-1) = G_{1,1}^{\,0,1} \!\left( \left. \begin{matrix} 1 \\ 0 \end{matrix} \; \right| \, x \right), \qquad \forall x </math>

Here, ''H'' denotes the [[Heaviside step function]].

The subsequent list shows how some [[special function|higher function]]s can be expressed in terms of the G-function:

:<math> \gamma (\alpha,x) = G_{1,2}^{\,1,1} \!\left( \left. \begin{matrix} 1 \\ \alpha,0 \end{matrix} \; \right| \, x \right), \qquad \forall x </math>

:<math> \Gamma (\alpha,x) = G_{1,2}^{\,2,0} \!\left( \left. \begin{matrix} 1 \\ \alpha,0 \end{matrix} \; \right| \, x \right), \qquad \forall x </math>

:<math> J_\nu (x) = G_{0,2}^{\,1,0} \!\left( \left. \begin{matrix} - \\ \frac{\nu}{2}, \frac{-\nu}{2} \end{matrix} \; \right| \, \frac{x^2}{4} \right), \qquad \frac{-\pi}{2} < \arg x \leq \frac{\pi}{2} </math>

:<math> Y_\nu (x) = G_{1,3}^{\,2,0} \!\left( \left. \begin{matrix} \frac{- \nu - 1}{2} \\ \frac{\nu}{2}, \frac{-\nu}{2}, \frac{- \nu - 1}{2} \end{matrix} \; \right| \, \frac{x^2}{4} \right), \qquad \frac{-\pi}{2} < \arg x \leq \frac{\pi}{2} </math>

:<math> I_\nu (x) = i^{-\nu} \; G_{0,2}^{\,1,0} \!\left( \left. \begin{matrix} - \\ \frac{\nu}{2}, \frac{-\nu}{2} \end{matrix} \; \right| \, -\frac{x^2}{4} \right), \qquad -\pi < \arg x \leq 0 </math>

:<math> K_\nu (x) = \frac{1}{2} \; G_{0,2}^{\,2,0} \!\left( \left. \begin{matrix} - \\ \frac{\nu}{2}, \frac{-\nu}{2} \end{matrix} \; \right| \, \frac{x^2}{4} \right), \qquad \frac{-\pi}{2} < \arg x \leq \frac{\pi}{2} </math>

:<math> \Phi (x,n,a) = G_{n+1,\,n+1}^{\,1,\,n+1} \!\left( \left. \begin{matrix} 0, 1-a, \dots, 1-a \\ 0, -a, \dots, -a \end{matrix} \; \right| \, -x \right), \qquad \forall x, \; n = 0,1,2,\dots </math>

:<math> \Phi (x,-n,a) = G_{n+1,\,n+1}^{\,1,\,n+1} \!\left( \left. \begin{matrix} 0, -a, \dots, -a \\ 0, 1-a, \dots, 1-a \end{matrix} \; \right| \, -x \right), \qquad \forall x, \; n = 0,1,2,\dots </math>

Even the [[incomplete gamma function#Derivatives|derivatives of γ(''α'',''x'') and Γ(''α'',''x'') with respect to ''α'']] can be expressed in terms of the Meijer G-function. Here, γ and Γ are the lower and upper [[incomplete gamma function]]s, ''J''<sub>''ν''</sub> and ''Y''<sub>''ν''</sub> are the [[Bessel function]]s of the first and second kind, respectively, ''I''<sub>''ν''</sub> and ''K''<sub>''ν''</sub> are the corresponding modified Bessel functions, and Φ is the [[Lerch transcendent]].

== See also==
*[[Gradshteyn and Ryzhik]]

==References==
{{Reflist}}
* {{cite book | last= Andrews | first= L. C. | title= Special Functions for Engineers and Applied Mathematicians | location= New York | publisher= MacMillan | year= 1985 | isbn= 978-0-02-948650-4 }}
* {{dlmf | id= 16.17 | first= R. A. | last= Askey | authorlink1=Richard Askey | first2= Adri B. Olde | last2= Daalhuis}}
* {{cite book | last1= Bateman | first1= H. | author1-link= Harry Bateman | last2= Erdélyi | first2= A. | author2-link= Arthur Erdélyi | title= Higher Transcendental Functions, Vol. I | url= http://apps.nrbook.com/bateman/Vol1.pdf | location= New York | publisher= McGraw–Hill | year= 1953 }} (see § 5.3, "Definition of the G-Function", p.&nbsp;206)
* {{cite journal | last1= Beals | first1= Richard  | last2= Szmigielski | first2= Jacek  | title= Meijer G-Functions: A Gentle Introduction | url= https://www.ams.org/notices/201307/rnoti-p866.pdf |  journal= Notices of the American Mathematical Society | volume= 60 | issue= 7  | pages= 866  | year= 2013 | doi= 10.1090/noti1016  | doi-access= free }} 
* {{eom | id= Meijer_transform&oldid=12567 | title= Meijer transform | first= Yu. A. | last= Brychkov | first2= A. P. | last2= Prudnikov}}
*{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor-first1=Daniel |editor-last1=Zwillinger |editor-first2=Victor Hugo |editor-last2=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=[[Academic Press, Inc.]] |date=2015 |orig-year=October 2014 |edition=8 |language=en |isbn=978-0-12-384933-5 |lccn=2014010276 <!-- |url=https://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21-->|title-link=Gradshteyn and Ryzhik |chapter=9.3.}}
* {{eom | id= Meijer-G-functions&oldid=13688 | title= Meijer G-functions | first= A. U. | last= Klimyk}}
* {{cite book | last= Luke | first= Yudell L. | author-link=Yudell Luke | title= The Special Functions and Their Approximations, Vol. I | location= New York | publisher= Academic Press | year= 1969 | isbn= 978-0-12-459901-7 }} (see Chapter V, "The Generalized Hypergeometric Function and the G-Function", p.&nbsp;136)
* {{cite journal | last= Meijer | first= C. S. | author-link= Cornelis Simon Meijer | title= Über Whittakersche bzw. Besselsche Funktionen und deren Produkte | language= de | journal= Nieuw Archief voor Wiskunde (2) | volume= 18 | issue= 4 | pages= 10&ndash;39 | year= 1936 | jfm= 62.0421.02 }}
* {{cite journal | last= Meijer | first= C. S. | title= Über eine Erweiterung der Laplace-Transformation &ndash; I, II | language= de | journal= Proceedings of the Section of Sciences, Koninklijke Akademie van Wetenschappen (Amsterdam) | volume= 43 | pages= 599&ndash;608 and 702&ndash;711 | year= 1940 | jfm= 66.0523.01 }}
* {{cite journal | last= Meijer | first= C. S. | title= Eine neue Erweiterung der Laplace-Transformation &ndash; I, II | language= de | journal= Proceedings of the Section of Sciences, Koninklijke Akademie van Wetenschappen (Amsterdam) | volume= 44 | pages= 727&ndash;737 and 831&ndash;839 | year= 1941a | jfm= 67.0396.01 }}
* {{cite journal | last= Meijer | first= C. S. | title= Multiplikationstheoreme für die Funktion <math> \scriptstyle G_{p,q}^{\,m,n}(z) </math> | language= de | journal= Proceedings of the Section of Sciences, Koninklijke Akademie van Wetenschappen (Amsterdam) | volume= 44 | pages= 1062&ndash;1070 | year= 1941b | jfm= 67.1016.01 }}
* {{cite journal | last= Narain | first= Roop | title= The G-functions as unsymmetrical Fourier kernels &ndash; I | url= https://www.ams.org/journals/proc/1962-013-06/S0002-9939-1962-0144157-5/S0002-9939-1962-0144157-5.pdf | journal= Proceedings of the American Mathematical Society | volume= 13 | issue= 6 | pages= 950&ndash;959 | year= 1962 | doi= 10.1090/S0002-9939-1962-0144157-5 | mr= 0144157 | doi-access= free }}
* {{cite journal | last= Narain | first= Roop | title= The G-functions as unsymmetrical Fourier kernels &ndash; II | url= https://www.ams.org/journals/proc/1963-014-01/S0002-9939-1963-0145263-2/S0002-9939-1963-0145263-2.pdf | journal= Proceedings of the American Mathematical Society | volume= 14 | issue= 1 | pages= 18&ndash;28 | year= 1963a | doi= 10.1090/S0002-9939-1963-0145263-2 | mr= 0145263 | doi-access= free }}
* {{cite journal | last= Narain | first= Roop | title= The G-functions as unsymmetrical Fourier kernels &ndash; III | url= https://www.ams.org/journals/proc/1963-014-02/S0002-9939-1963-0149210-9/S0002-9939-1963-0149210-9.pdf | journal= Proceedings of the American Mathematical Society | volume= 14 | issue= 2 | pages= 271&ndash;277 | year= 1963b | doi= 10.1090/S0002-9939-1963-0149210-9 | mr= 0149210 | doi-access= free }}
* {{cite book | last1= Prudnikov | first1= A. P. | last2= Marichev|first2= O. I.|last3 = Brychkov | first3= Yu. A. | title= Integrals and Series, Vol. 3: More Special Functions | location= Newark, NJ | publisher= Gordon and Breach | year= 1990 | isbn= 978-2-88124-682-1 }} (see § 8.2, "The Meijer G-function", p.&nbsp;617)
* {{cite book | last= Slater | first= Lucy Joan | author-link= Lucy Joan Slater | title= Generalized hypergeometric functions | url= https://archive.org/details/generalizedhyper0000unse | url-access= registration | location= Cambridge, UK | publisher= Cambridge University Press | year= 1966 | isbn= 978-0-521-06483-5 }} (there is a 2008 paperback with {{isbn|978-0-521-09061-2}})
* {{cite journal | last= Wimp | first= Jet | title= A Class of Integral Transforms | journal= Proceedings of the Edinburgh Mathematical Society |series=Series 2 | volume= 14 | pages= 33&ndash;40 | year= 1964 | doi= 10.1017/S0013091500011202 | mr= 0164204 | zbl= 0127.05701 | doi-access= free }}
*{{Cite book |last=Mathai, Saxena |first=A.M and R.K |title=Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences |publisher=[[Springer Publishing|Springer]] |year=1973 |isbn=9780387064826}}
* {{cite journal |last1=Good |first1=Michael |title=Radiation from an inertial mirror horizon |journal=Universe |date=2020 |volume=6(9) |issue=131 |page=131 |doi=10.3390/universe6090131 |arxiv=2008.08776 |bibcode=2020Univ....6..131G |doi-access=free }}

==External links==
* {{mathworld | urlname= MeijerG-Function | title= Meijer G-Function}}
* [https://gitlab.com/RZ-FZJ/hypergeom hypergeom] on [[GitLab]]

[[Category:Hypergeometric functions]]