Meijer G-Function

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 hypergeometric function and MacRobert E-function had the same aim, but Meijer's G-function was able to include those as a particular case as well. The first definition was made by Meijer using a series; nowadays the accepted and more general definition is via an integral on the complex plane, introduced firstly by 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.

Definition

In general the G-function is defined with the following integral on the complex plane:

:

The function is defined under the following hypothesis:
* $0\; leq\; m\; leq\; q$, $0\; leq\; n\; leq\; p$ and $p\; leq\; q-1$
* $z\; eq\; 0$
* no couple of $b\_k,\; (k\; =\; 1,2,dots,m)$ differs by an integer or a zero
* the parameters $a\_h$ and $b\_h$ are so that no pole of $Gamma\; (b\_j\; -\; s),\; j\; =\; 1,2,dots,m$ coincide with any pole of $Gamma\; (1\; -\; a\_k\; +\; s),\; k\; =\; 1,2,dots,n$
* $a\_j\; -\; b\_k\; eq\; 1,2,3,dots$ for $j\; =\; 1,2,dots,n$ and $k\; =\; 1,2,dots,m$
* if $p\; =\; q$, then the definition makes sense only for

The G-function is an analytic function of "z" with a discontinuity in the origin. It is common to use the following more synthetic notation using vectors:

:

the "L" in the integral represents the path to follow while integrating. There are three possible paths:

:1. "L" goes from $-\; i\; infty$ to $+\; i\; infty$ so 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 of the path. So that the integral converges, it is necessary that $delta\; =\; m\; +\; n\; -\; frac\{1\}\{2\}\; (p+q)>\; 0$, . Defining:::$u\; =\; sum\_\{j\; =\; 1\}^\{q\}\; b\_j\; -\; sum\_\{j\; =\; 1\}^\{p\}\; a\_j$:when $|\; arg\; z\; |\; =\; delta\; pi,\; delta\; geq\; 0$:
* when $p\; =\; q$, the integral converges if
* when $p\; eq\; q$, expressing $s\; =\; sigma\; +\; i\; au$, where $sigma$ and $au$ are real, the integral converges if, when $au\; o\; infty$, the following relationship is true:::$(q\; -\; p)\; sigma\; >\; mbox\{Re\; \}\{\; u\}\; +\; 1\; -\; frac\{1\}\{2\}$:2. "L" is a loop beginning and ending at $+infty$, encircling all poles of $Gamma\; (b\_j\; -\; s),\; j\; =\; 1,2,dots,m$ only once in the negative direction, but not encircling any pole of $Gamma\; (1\; -\; a\_k\; +\; s),\; k\; =\; 1,2,dots,n$. The integral converges if $q\; geq\; 1$ and $p\; leq\; q$; in the particular case of $p\; =\; q$ it must be .:3. "L" is a loop beginning and ending at $-infty$ and encircling all poles of $Gamma\; (1\; -\; a\_k\; +\; s),\; k\; =\; 1,2,dots,n$, once in the positive direction, but not encircling any pole of $Gamma\; (b\_j\; -\; s),\; j\; =\; 1,2,dots,m$. The integral converges if $p\; geq\; 1$ and $p\; geq\; q$; in the particular case of $p\; =\; q$ it must be $|z|>1$.

It is possible to show that, if the integral converges for more than one of these three paths, then the result is the same. If the integral converges for only one path, then that is the only one to be considered.

If the integral converges if calculated along the second path, then the G-function can be expressed as a sum of residues, also using the Generalized hypergeometric function:

:

This relationship is valid only when the integral converges among the second path, i.e. when , or when $p\; =\; q$ and . The asterisks have a particular meaning. In the product it reminds to ignore the case $b\_j\; =\; b\_h$, replacing $Gamma(0)$ with $1$. In the other case, in the argument of the hypergeometric function, remembering the meaning of the vector notation:

:$1\; +\; b\_h\; -\; mathbf\{b\_q\}\; =\; (1\; +\; b\_h\; -\; b\_1)\; cdots\; (1\; +\; b\_h\; -\; b\_i)\; cdots\; (1\; +\; b\_h\; -\; b\_q)$

the asterisk reminds to ignore the case $b\_i\; =\; b\_h$, shortening the vector length from $q$ to $q-1$.

When $m\; =\; 0$, the second path does not contain any pole, so the value of the integral is always zero:

:

From these considerations we can understand how the G-function is a further generalization of the generalized hypergeometric function. The G-function is defined for any value of "p" and "q", but in the particular case when the integral is defined among the second path, then the G-function can be expressed in terms of the hypergeometric function. In other terms, introducing the G-function we can find a solutions for the differential equation of the hypergeometric function for $p\; >\; q\; +\; 1$ as well.

The differential equation of the G-function

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

:$left\; [\; (-1)^\{m\; +\; n\; -\; p\}\; z\; prod\_\{j\; =\; 1\}^\{p\}\; left(\; z\; frac\{d\}\{dz\}\; -\; a\_j\; +\; 1\; ight)\; -\; prod\_\{i\; =\; 1\}^\{q\}\; left(\; z\; frac\{d\}\{dz\}\; -\; b\_i\; ight)\; ight]\; U(z)\; =\; 0.$

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

Analytic continuation of the G-function

The following property of the G-function is called "analytic continuation", it is possible to show that from the definition:

:

This property is really important: using it we can transform a G-function with $p>q$ in another one with

Relationship between G-function and hypergeometric function

The hypergeometric function can always be expressed in terms of the G-function:

:

where we have used the vector notation:

:$Gamma(mathbf\{a\_p\})\; =\; prod\_\{j\; =\; 1\}^\{p\}\; Gamma(a\_j)$

using the "analytic continuation" property, it is possible to express it in a sightly different form:

:

both relationships are valid if $;\_p\; F\_q\; (cdot)$ is defined, i. e. $p\; leq\; q$ or $p\; =\; q+1$ with .

Elementary properties of the G-function

As is clear from the definition, the factors $mathbf\{a\_p\}$ and $mathbf\{b\_q\}$ are on the numerator and on the denominator of a fraction; that is why, if there are equal parameters, it is possible to simplify them, thus reducing the order of the function.Whether it will be "m" or "n" to decrease, this depends of the position of a factor compared to the other. As an example, if one of $a\_h,\; h\; =\; 1,2,dots,\; n$ equals one of $b\_j,\; j\; =\; m+1,dots,\; q$, the G-function lowers its order:

:

For the same reason, if one of $a\_h,\; h\; =\; n+1,dots,\; p$ equals one of $b\_j,\; j\; =\; 1,2,dots,\; m$, then:

:

Moreover, starting from the definition, it is possible to prove the following relationships:

:

:

About derivatives, there are the following relationships:

:

:

:

:

From these four properties, it is possible to deduce others simply calculating the derivative on the left of the equal and manipulating a bit. For example:

:

Moreover:

:

:

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

Multiplication theorem

Provided that $z\; eq\; 0$, and that "m", "n", "p" and "q" are integer with:$q\; geq\; 1,\; qquad0\; leq\; n\; leq\; p\; leq\; q,\; qquad0\; leq\; m\; leq\; q$the following relationship is valid::

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.

Integrals involving G-function

There is the following relationship for integrating the G-function:

:

This relationship is valid provided that $p\; leq\; q$; if $p\; >\; q$ we can use the "analytic continuation" property:

::

It is possible to represent the integral of a product of two G-functions with just one function:

:::

Laplace transform

Using the previous relationships it is possible to prove that:

:

if we put $alpha\; =\; 0$ 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:

:

where "c" is a real positive constant, "z" is real and $z,y\; eq\; 0$.

This is another Laplace transform involving the G-function:

:

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:

:$g(z)\; =\; int\_\{0\}^\{infty\}\; k(z,y)\; f(y)\; dy,$:$f(z)\; =\; int\_\{0\}^\{infty\}\; h(z,y)\; g(y)\; dy$

are both verified 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:

:

:

are two asymmetric kernels. In particular, if $p\; =\; q$, $m\; =\; n$, $a\_j\; +\; b\_j\; =\; 0$ for $j\; =\; 1,\; 2,\; dots,\; p$ and $b\_h\; +\; d\_h\; =\; 0$ for $h\; =\; 1,\; 2,\; dots,\; m$, then the two kernels become symmetric.

Wimp transform

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

:

:$h(z,y)\; =\; frac\{i\}\{pi\}\; y\; e^\{-\; u\; pi\; i\}\; left\; [\; e^\{pi\; y\}\; A(\; u\; +\; i\; y,\; u\; -\; i\; y|z\; e^\{i\; pi\}\; )\; -\; e^\{-\; pi\; y\}\; A(\; u\; -\; i\; y,\; u\; +\; i\; y\; |\; z\; e^\{i\; pi\}\; )\; ight]$

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

:

Relationship between the G-function and other elementary functions

The following list shows how it is possible to express several functions in terms of the G-function:

::::::::::

The last two functions are the Bessel functions of the first and second kind.

References

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