Meijer G-function: Information from Answers.com

The G-function was defined for the first time by the Dutch mathematician Cornelis Simon Meijer (1904–1974) in 1936 as an attempt to introduce a very general function that includes most of the known special functions as particular cases. This was not the only attempt: 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 path integral in the complex plane, introduced firstly by Arthur Erdélyi in 1953. With the current definition, it is possible to express most of the special functions in terms of the G-function and of the Gamma function.

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

1 Definition of the Meijer G-function
- 1.1 Differential equation
2 Relationship between the G-function and the generalized hypergeometric function
3 Basic properties of the G-function
- 3.1 Multiplication theorem
4 Definite integrals involving the G-function
5 Laplace transform of the G-function
6 Integral transforms using the G-function
- 6.1 Narain transform
- 6.2 Wimp transform
7 Representation of other functions in terms of the G-function
8 References

Definition of the Meijer G-function

A general definition of the Meijer G-function is given by the following path integral in the complex plane:

$G_{p,q}^{\,m,n} \left( \left. \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix}\; \right| \; z \right) = \frac{1}{2 \pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s) \prod_{j=1}^{n}\Gamma(1 - a_j +s)} {\prod_{j=m+1}^{q} \Gamma(1 - b_j + s) \prod_{j=n+1}^{p}\Gamma(a_j - s)} z^s ds ~.$

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:

$0 \leq m \leq q$ and $0 \leq n \leq p$ , where m, n, p and q are integer numbers
$a_k - b_j \neq 1,2,3,\dots$ for $k = 1,2,\dots,n$ and $j = 1,2,\dots,m$ , which implies that no pole of any $\Gamma (b_j - s), ~j = 1,2,\dots,m$ coincides with any pole of any $\Gamma (1 - a_k + s), ~k = 1,2,\dots,n$
$z \neq 0$

The G-function is an analytic function of z with possible exception of the origin $z = 0$ and of the unit circle $| z | = 1$ . One often encounters the following more synthetic notation using vectors:

$G_{p,q}^{\,m,n} \left(\left. \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \; \right| \; z \right) = G_{p,q}^{\,m,n} \left(\left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; z \right).$

The L in the integral represents the path to be followed while integrating. Three choices are possible for this path:

1. L goes from $- i \infty$ to $+ i \infty$ such that all poles of $\Gamma (b_j - s), ~j = 1,2,\dots,m$ are on the right of the path, while all poles of $\Gamma (1 - a_k + s), ~k = 1,2,\dots,n$ are on the left. The integral then converges for $| \arg z | < \delta \pi$ , where

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

an obvious prerequisite for this is

δ > 0

. The integral additionally converges for $| \arg z | = \delta \pi \geq 0$ if

$(q - p) (\sigma + \frac{1}{2}) > \mbox{Re}\{\nu\} + 1 ~,$

where

σ

represents

Re{s}

as the integration variable s approaches both $+ i \infty$ and $- i \infty$ , and where

$\nu = \sum_{j = 1}^{q} b_j - \sum_{j = 1}^{p} a_j ~.$

As a corollary, for

| arg z | = δπ

and

p = q

the integral converges independent of

σ

whenever

Re{ν} < - 1

2. L is a loop beginning and ending at $+\infty$ , encircling all poles of $\Gamma (b_j - s), ~j = 1,2,\dots,m$ exactly once in the negative direction, but not encircling any pole of $\Gamma (1 - a_k + s), ~k = 1,2,\dots,n$ . Then the integral converges for all z if $q > p \geq 0$ ; it also converges for

q = p > 0

as long as

| z | < 1

. In the latter case, the integral additionally converges for

| z | = 1

Re{ν} < - 1

, where

ν

is defined as for the first path.

3. L is a loop beginning and ending at $-\infty$ and encircling all poles of $\Gamma (1 - a_k + s), ~k = 1,2,\dots,n$ , exactly once in the positive direction, but not encircling any pole of $\Gamma (b_j - s), ~j = 1,2,\dots,m$ . Now the integral converges for all z if $p > q \geq 0$ ; it also converges for

p = q > 0

as long as

| z | > 1

. As already stated for the second path, in the case of

p = q

the integral also converges for

| z | = 1

when

Re{ν} < - 1

The conditions for convergence are easily obtained by applying Stirling's asymptotic approximation to the $Γ$ functions in the integrand. It is possible to show that, if the integral converges for more than one of these three paths, then the results are the same. If the integral 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.

Differential equation

The G-function is the solution of the following differential equation:

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

The order of the equation is $max(p, q)$ .

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 pair among the $b_j, ~j = 1,2,\dots,m$ differs by an integer or zero, then the Meijer G-function can be expressed as a sum of residues in terms of Generalized hypergeometric functions $p F q - 1$ (Slater's theorem):

$G_{p,q}^{\,m,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; z \right) = \sum_{h=1}^{m} \frac{\prod_{j=1}^m \Gamma(b_j - b_h)^* \prod_{j=1}^{n}\Gamma(1+b_h - a_j) \;z^{b_h}} {\prod_{j=m+1}^{q} \Gamma(1+b_h - b_j) \prod_{j=n+1}^{p}\Gamma(a_j - b_h)} \times$

$\times \;_{p}F_{q-1} \left( \left. \begin{matrix} 1+b_h - \mathbf{a_p} \\ (1+b_h - \mathbf{b_q})^* \end{matrix} \; \right| \; (-1)^{p-m-n} \;z \right).$

For the integral to converge along the second path one must have either $p < q$ , or $p = q$ and $| z | < 1$ . 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,

$1 + b_h - \mathbf{b_q} = (1 + b_h - b_1) \cdots (1 + b_h - b_j) \cdots (1 + b_h - b_q) ~,$

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,

$G_{p,q}^{\,0,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; z\right) = 0 ~,$

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 pair among the $a_k, ~k = 1,2,\dots,n$ differs by an integer or zero, then the G-function can be expressed as:

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

$\times \;_{q}F_{p-1} \left( \left. \begin{matrix} 1-a_h + \mathbf{b_q} \\ (1-a_h + \mathbf{a_p})^* \end{matrix} \; \right| \; \frac{(-1)^{q-m-n}}{z} \right).$

For $n = 0$ one consequently has:

$G_{p,q}^{\,m,0} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; z\right) = 0 ~.$

Here, either $q > p$ , or $q = p$ and $| z | > 1$ are required.

On the other hand, any generalized hypergeometric function can readily be expressed in terms of the Meijer G-function:

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

where we have made use of the vector notation:

$\Gamma(\mathbf{a_p}) = \prod_{j = 1}^{p} \Gamma(a_j)~.$

This relationship is valid whenever the generalized hypergeometric series $p F q (z)$ converges, i. e. for any finite z when $p \leq q$ , and for $| z | < 1$ when $p = q + 1$ . In the latter case, the relation with the G-function automatically provides the analytic continuation of $p F q (z)$ to $|z| \geq 1$ with a branch cut from 1 to $\infty$ 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.

Basic properties of the G-function

As can be seen from the definition of the G-function, if equal parameters appear among the $\mathbf{a_p}$ and $\mathbf{b_q}$ determining the factors in the numerator and the denominator of the integrand, it is possible to simplify the fraction, thus reducing the order of the function. Whether the parameter m or n will decrease depends of the particular position of the factors in question. For instance, if one of the $a_k, ~k = 1,2,\dots, n$ equals one of the $b_j, ~j = m+1,\dots, q$ , the G-function lowers its orders p, q and n:

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

For the same reason, if one of the $a_k, ~k = n+1,\dots, p$ equals one of the $b_j, ~j = 1,2,\dots, m$ , then:

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

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

$z^{\alpha} \;G_{p,q}^{\,m,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; z \right) = G_{p,q}^{\,m,n} \left( \left. \begin{matrix} \mathbf{a_p} + \alpha \\ \mathbf{b_q} + \alpha \end{matrix} \; \right| \; z \right),$

$G_{p+1,\,q+1}^{\,m,\,n+1} \left( \left. \begin{matrix} \alpha, \mathbf{a_p} \\ \mathbf{b_q}, \beta \end{matrix} \; \right| \; z \right) = (-1)^{\beta-\alpha} \;G_{p+1,\,q+1}^{\,m+1,\,n} \left( \left. \begin{matrix} \mathbf{a_p}, \alpha \\ \beta, \mathbf{b_q} \end{matrix} \; \right| \; z \right), \quad q \geq m, \; \beta-\alpha = 0,1,2,\dots ~,$

$G_{p,q}^{\,m,n} \left(\left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; z \right) = G_{q,p}^{\,n,m} \left( \left. \begin{matrix} 1-\mathbf{b_q} \\ 1-\mathbf{a_p} \end{matrix} \; \right| \; \frac{1}{z} \right),$

$G_{p,q}^{\,m,n} \left(\left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; z \right) = \frac{k^{1+\nu+(p-q)/2}} {(2 \pi)^{(k-1) \delta}} G_{k p, \, k q}^{\, k m, \, k n} \left( \left. \begin{matrix} a_1/k, \dots, (a_1+k-1)/k, \dots, a_p/k, \dots, (a_p+k-1)/k \\ b_1/k, \dots, (b_1+k-1)/k, \dots, b_q/k, \dots, (b_q+k-1)/k \end{matrix} \; \right| \; \frac{z^k} {k^{k(q-p)}} \right), \quad k = 1,2,3,\dots ~.$

The abbreviations $ν$ and $δ$ were introduced in the definition of the G-function above.

Concerning derivatives, there are these relationships:

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

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

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

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

From these four, it is possible to deduce other relations simply by calculating the derivative on the left-hand side and manipulating a bit. One obtains for example:

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

Moreover, for derivatives of arbitrary order k, one has

$z^k \frac{d^k}{dz^k} G_{p,q}^{\,m,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; z \right) = G_{p+1,\,q+1}^{\,m,\,n+1} \left( \left. \begin{matrix} 0, \mathbf{a_p} \\ \mathbf{b_q}, k \end{matrix} \; \right| \; z \right),$

$z^k \frac{d^k}{dz^k} G_{p,q}^{\,m,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; \frac{1}{z} \right) = G_{p+1,\,q+1}^{\,m+1,\,n} \left( \left. \begin{matrix}\mathbf{a_p}, 1-k \\ 1, \mathbf{b_q} \end{matrix} \; \right| \; \frac{1}{z} \right),$

which hold for $k < 0$ as well, thus allowing to obtain the antiderivative of any G-function, with one caveat: the set of parameters in the result must not violate the requirement $a_k - b_j \neq 1,2,3,\dots$ for $k = 1,2,\dots,n$ and $j = 1,2,\dots,m$ from the definition above.

Several properties of the hypergeometric function and of other special functions can be deduced from these relationships.

Multiplication theorem

Provided that $z \neq 0$ , and that for the integer parameters m, n, p and q

$q \geq 1 ~, \qquad 0 \leq n \leq p \leq q ~, \qquad 0 \leq m \leq q ~,$

the following relationship holds:

$G_{p,q}^{\,m,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; w z \right) = \sum_{k=0}^{\infty} \frac{(w - 1)^k}{k!} G_{p+1,\,q +1}^{\,m,\,n+1} \left( \left. \begin{matrix} 0, \mathbf{a_p} \\ \mathbf{b_q},k \end{matrix} \; \right| \; z \right).$

It is possible to prove this using the elementary properties discussed above. This theorem is the generalization of similar theorems for Bessel and hypergeometric functions.

Definite integrals involving the G-function

Among definite integrals involving an arbitrary G-function one has:

$\int_0^{\infty} z^{s - 1} \;G_{p,q}^{\,m,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; \eta z \right) dz = \frac{\eta^{-s} \prod_{j = 1}^{m} \Gamma (b_j + s) \prod_{j = 1}^{n} \Gamma (1 - a_j - s)} {\prod_{j = m + 1}^{q} \Gamma (1 - b_j - s) \prod_{j = n + 1}^{p} \Gamma (a_j + s)} ~.$

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 above.

A result of fundamental importance is that the definite integral of a product of two arbitrary G-functions can be represented by just another G-function (convolution theorem):

$\int_0^{\infty} G_{p,q}^{\,m,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; \eta x \right) G_{\sigma, \tau}^{\,\mu, \nu} \left( \left. \begin{matrix} \mathbf{c_{\sigma}} \\ \mathbf{d_\tau} \end{matrix} \; \right| \; \omega x \right) dx =$

$= \frac{1}{\eta} G_{q + \sigma ,\, p + \tau}^{\,n + \mu ,\, m + \nu} \left( \left. \begin{matrix} - b_1, \dots, - b_m, \mathbf{c_{\sigma}}, - b_{m+1}, \dots, - b_q \\ - a_1, \dots, -a_n, \mathbf{d_\tau} , - a_{n+1}, \dots, - a_p \end{matrix} \; \right| \; \frac{\omega}{\eta} \right) =$

$= \frac{1}{\omega} G_{p + \tau ,\, q + \sigma}^{\,m + \nu ,\, n + \mu} \left( \left. \begin{matrix} a_1, \dots, a_n, -\mathbf{d_\tau} , a_{n+1}, \dots, a_p \\ b_1, \dots, b_m, -\mathbf{c_{\sigma}}, b_{m+1}, \dots, b_q \end{matrix} \; \right| \; \frac{\eta}{\omega} \right).$

Again, the restrictions under which the integral exists have been omitted here. Note how the Mellin transform of the result merely assembles the $Γ$ factors from the Mellin transforms of the two functions in the integrand. Many of the amazing definite integrals listed in tables or produced by Computer Algebra Systems are nothing but special cases of this formula.

Laplace transform of the G-function

Using the previous relationships it is possible to prove that:

$\int_0^{\infty} e^{- \omega y} \;y^{- \alpha} \;G_{p,q}^{\,m,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; z y \right) dy = \omega^{\alpha - 1} \;G_{p + 1,\,q}^{\,m,\,n+1} \left( \left. \begin{matrix} \alpha, \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; \frac{z}{\omega} \right).$

If we put $α = 0$ we get the Laplace transform of the G-function, so we can view this relationship as a generalized Laplace transform. The inverse is given by:

$z^{- \alpha} \;G_{p,\,q+1}^{\,m,\,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q}, \alpha \end{matrix} \; \right| \; z y \right) = \frac{1}{2 \pi i} \int_{c - i \infty}^{c + i \infty} e^{\omega z} \;\omega^{\alpha - 1} \;G_{p,q}^{\,m,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; \frac{y}{\omega} \right) d\omega ~,$

where c is a real positive constant, z is real and $z,y \neq 0$ .

Another Laplace transform involving the G-function is:

$\int_{0}^{\infty} e^{- \beta x} \;G_{p,q}^{\,m,n} \left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; \alpha x^2 \right) dx = \frac{1}{\sqrt{\pi} \beta} G_{p+2,\,q}^{\,m,\,n+2} \left( \left. \begin{matrix} 0,\frac{1}{2},\mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \; \frac{4 \alpha}{\beta^2} \right).$

The restrictions under which the integrals exist have been omitted in both cases.

Integral transforms using the G-function

In general, two functions $k (z, y)$ and $h (z, y)$ are called transform kernels if, for any two functions $f (z)$ and $g (z)$ , these two relationships:

$g(z) = \int_{0}^{\infty} k(z,y) f(y) dy ~,$

$f(z) = \int_{0}^{\infty} h(z,y) g(y) dy$

both hold at the same time. The two kernels are said to be symmetric if $k (z, y) = h (z, y)$ .

Narain transform

Narain (1962, 1963) showed that the functions:

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

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

are two asymmetric kernels. In particular, if $p = q$ , $m = n$ , $\;a_j + b_j = 0$ for $j = 1, 2, \dots, p$ and $c j - d j = 0$ for $j = 1, 2, \dots, m$ , then the two kernels become symmetric.

Wimp transform

Wimp (1964) showed that these two functions are asymmetric transform kernels:

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

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

where the function $A(\cdot)$ is defined as:

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

Representation of other functions in terms of the G-function

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

$e^x = G_{0,1}^{\,1,0} \left( \left. \begin{matrix} - \\ 0 \end{matrix} \; \right| \; -x \right) , \qquad \forall x$

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

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

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

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

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

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

$\ln (1+x) = G_{2,2}^{\,1,2} \left( \left. \begin{matrix} 1,1 \\ 1,0 \end{matrix} \; \right| \; x \right) , \qquad \forall x$

The following list shows how some higher functions can be expressed in terms of the G-function:

$\gamma (\alpha,x) = G_{1,2}^{\,1,1} \left( \left. \begin{matrix} 1 \\ \alpha,0 \end{matrix} \; \right| \; x \right) , \qquad \forall x$

$\Gamma (\alpha,x) = G_{1,2}^{\,2,0} \left( \left. \begin{matrix} 1 \\ \alpha,0 \end{matrix} \; \right| \; x \right) , \qquad \forall x$

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

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

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

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

$\Phi (x,s,a) = G_{s+1,\,s+1}^{\,1,\,s+1} \left( \left. \begin{matrix} 0, 1-a, \dots, 1-a \\ 0, -a, \dots, -a \end{matrix} \; \right| \; -x \right) , \qquad \forall x, \; s = 0,1,2,\dots$

Here, $γ$ and $Γ$ are the lower and upper incomplete gamma functions, $J ν$ and $Y ν$ are the Bessel functions of the first and second kind, respectively, $I ν$ and $K ν$ are the corresponding modified Bessel functions, and $Φ$ is the Lerch transcendent.

References

C. S. Meijer, "Über Whittakersche bezw. Besselsche Funktionen und deren Produkte", Nieuw Archief voor Wiskunde, 18, No 4 (1936), pp. 10-39.
Luke, Y. L. (1969). The Special Functions and Their Approximations, Volume I. New York: Academic Press.
Andrews, L. C. (1985). Special Functions for Engineers and Applied Mathematicians. New York: MacMillan.

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Meijer G-function

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