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 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 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 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 $0\leq n\leq p$
• $a_{k}-b_{j}\neq 1,2,3,\dots$ for $k=1,2,\dots ,n$ and $j=1,2,\dots ,m$, which means that no pole of $\Gamma (b_{j}-s),j=1,2,\dots ,m$ coincides with any pole of $\Gamma (1-a_{k}+s),k=1,2,\dots ,n$
• $z\neq 0$

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 follow 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

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

obviously, $\delta >0$ is a prerequisite for this. The integral additionally converges for $|\arg z|=\delta \pi \geq 0$ if

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

where $\sigma ={\mbox{Re}}\{s\}$ as the integration variable s approaces $+i\infty$ as well as $-i\infty$, and where

$\nu =\sum _{j=1}^{q}b_{j}-\sum _{j=1}^{p}a_{j}~.$

As a corollary, when $p=q$ the integral converges independent of $\sigma$ if ${\mbox{Re}}\{\nu \}<-1$.

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 if $q>p\geq 0$; it also converges for $q=p>0$ as long as $|z|<1$.

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 if $p>q\geq 0$; it also converges for $p=q>0$ as long as $|z|>1$.

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 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.

The differential equation of the G-function

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

$\left[(-1)^{m+n-p}z\prod _{j=1}^{p}\left(z{\frac {d}{dz}}-a_{j}+1\right)-\prod _{i=1}^{q}\left(z{\frac {d}{dz}}-b_{i}\right)\right]G(z)=0.$

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

Relationship between G-function and the generalized hypergeometric function

If the integral converges when evalulated along the second path introduced above, and if no couple of $b_{j},(j=1,2,\dots ,m)$ differs by an integer or zero, then the G-function can be expressed as a sum of residues in terms of Generalized hypergeometric functions $_{p}F_{q-1}$ (Slater's theorem):

$G_{p,q}^{m,n}\left(\left.{\begin{matrix}\mathbf {a_{p}} \\\mathbf {b_{q}} \end{matrix}}\;\right|\;z\right)=\sum _{h=1}^{m}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}-b_{h})^{*}\prod _{j=1}^{n}\Gamma (1+b_{h}-a_{j})z^{b_{h}}}{\prod _{j=m+1}^{q}\Gamma (1+b_{h}-b_{j})\prod _{j=n+1}^{p}\Gamma (a_{j}-b_{h})}}\times \;_{p}F_{q-1}\left(\left.{\begin{matrix}1+b_{h}-\mathbf {a_{p}} \\(1+b_{h}-\mathbf {b_{q}} )^{*}\end{matrix}}\;\right|\;(-1)^{p-m-n}z\right).$

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 are to be understood as follows: In the product the asterisk reminds to ignore the case $b_{j}=b_{h}$, which amounts to replacing $\Gamma (0)$ with $1$. In the argument of the hypergeometric function, on the other hand, if we recall the meaning of the vector notation

$1+b_{h}-\mathbf {b_{q}} =(1+b_{h}-b_{1})\cdots (1+b_{h}-b_{j})\cdots (1+b_{h}-b_{q})~,$

the asterisk reminds to ignore the case $b_{j}=b_{h}$, which amounts to shortening the vector length from $q$ to $q-1$.

Note that when $m=0$, the second path does not contain any pole, so the value of the integral must be always zero:

$G_{p,q}^{0,n}\left(\left.{\begin{matrix}\mathbf {a_{p}} \\\mathbf {b_{q}} \end{matrix}}\;\right|\;z\right)=0;\quad p\leq q.$

Similarly, if the integral converges when evalulated along the third path above, and if no couple of $a_{k},(k=1,2,\dots ,n)$ differs by an integer or zero, then the G-function can be expressed as:

$G_{p,q}^{m,n}\left(\left.{\begin{matrix}\mathbf {a_{p}} \\\mathbf {b_{q}} \end{matrix}}\;\right|\;z\right)=\sum _{h=1}^{n}{\frac {\prod _{j=1}^{m}\Gamma (a_{h}-a_{j})^{*}\prod _{j=1}^{m}\Gamma (1+b_{j}-a_{h})z^{a_{h}-1}}{\prod _{j=n+1}^{p}\Gamma (1+a_{j}-a_{h})\prod _{j=m+1}^{q}\Gamma (a_{h}-b_{j})}}\times \;_{q}F_{p-1}\left(\left.{\begin{matrix}1-a_{h}+\mathbf {b_{q}} \\(1-a_{h}+\mathbf {a_{p}} )^{*}\end{matrix}}\;\right|\;{\frac {(-1)^{q-m-n}}{z}}\right).$

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

On the other hand, the hypergeometric function can always be expressed in terms of the G-function:

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

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

where we have used the vector notation:

$\Gamma (\mathbf {a_{p}} )=\prod _{j=1}^{p}\Gamma (a_{j})$

Both relationships are valid if $\;_{p}F_{q}(\cdot )$ is defined, i. e. $p\leq q$ or $p=q+1$ with $0<|z|<1$.

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.

Basic properties of the G-function

As is clear from the definition, 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 of the G-function will decrease depends of the particular position of the factors in question. For instance, 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 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 $a_{h},h=n+1,\dots ,p$ equals one of $b_{j},j=1,2,\dots ,m$, then:

$G_{p,q}^{m,n}\left(\left.{\begin{matrix}a_{1},\dots ,a_{p-1},b_{1}\\b_{1},b_{2},\dots ,b_{q}\end{matrix}}\;\right|\;z\right)=G_{p-1,q-1}^{m-1,n}\left(\left.{\begin{matrix}a_{1},\dots ,a_{p-1}\\b_{2},\dots ,b_{q}\end{matrix}}\;\right|\;z\right),\quad m,p,q\geq 1$

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

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

$G_{p+1,q+1}^{m,n+1}\left(\left.{\begin{matrix}a,\mathbf {a_{p}} \\\mathbf {b_{q}} ,b\end{matrix}}\;\right|\;z\right)=(-1)^{a-b}G_{p+1,q+1}^{m+1,n}\left(\left.{\begin{matrix}\mathbf {a_{p}} ,a\\b,\mathbf {b_{q}} \end{matrix}}\;\right|\;z\right),\quad q\geq m,\;a-b=0,1,2,\dots$

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

About 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 relations, it is possible to deduce others simply by calculating the derivative on the left-hand side and manipulating a bit. 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, one has

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

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

which hold for $k<0$ as well, thus allowing to integrate any G-function – unless the parameters of the result 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 m, n, p and q are integer with

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

the following relationship is valid:

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

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 iuntegrals involving an arbitrary G-function one has:

$\int _{0}^{\infty }z^{s-1}G_{p,q}^{m,n}\left(\left.{\begin{matrix}\mathbf {a_{p}} \\\mathbf {b_{q}} \end{matrix}}\;\right|\;\eta z\right)dz={\frac {\eta ^{-s}\prod _{j=1}^{m}\Gamma (b_{j}+s)\prod _{j=1}^{n}\Gamma (1-a_{j}-s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}-s)\prod _{j=n+1}^{p}\Gamma (a_{j}+s)}}$

Noteb 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 of the definition given 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:

$\int _{0}^{\infty }G_{p,q}^{m,n}\left(\left.{\begin{matrix}\mathbf {a_{p}} \\\mathbf {b_{q}} \end{matrix}}\;\right|\;\eta x\right)G_{\sigma ,\tau }^{\mu ,\nu }\left(\left.{\begin{matrix}\mathbf {c_{\sigma }} \\\mathbf {d_{\tau }} \end{matrix}}\;\right|\;\omega x\right)dx=$

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

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

Again, the restrictions under which the integral exists have been omitted here. Many of the more amazing results found in tables of integrals, or produced by Computer Algebra Systems, are nothing but special cases of this convolution formula.

Laplace transform

Using the previous relationships it is possible to prove that:

$\int _{0}^{\infty }e^{-\omega y}y^{-\alpha }G_{p,q}^{m,n}\left(\left.{\begin{matrix}\mathbf {a_{p}} \\\mathbf {b_{q}} \end{matrix}}\;\right|\;zy\right)dy=\omega ^{\alpha -1}G_{p+1,q}^{m,n+1}\left(\left.{\begin{matrix}\alpha ,\mathbf {a_{p}} \\\mathbf {b_{q}} \end{matrix}}\;\right|\;{\frac {z}{\omega }}\right)$

if we put $\alpha =0$ 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:

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

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

This is another Laplace transform involving the G-function:

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

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:

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

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

are two asymmetric kernels. In particular, if $p=q$, $m=n$, $a_{j}+b_{j}=0$ for $j=1,2,\dots ,p$ and $b_{h}+d_{h}=0$ 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:

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

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

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:

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

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

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

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

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

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

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

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

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

$J_{\alpha }(x)=G_{0,2}^{1,0}\left(\left.{\begin{matrix}-\\{\frac {\alpha }{2}},{\frac {-\alpha }{2}}\end{matrix}}\;\right|\;{\frac {x^{2}}{4}}\right),\qquad x\geq 0$

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

Here, $J_{\alpha }$ and $Y_{\alpha }$ are the Bessel functions of the first and second kind, respectively.

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.