Meijer G-Function

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The G-function was defined for the first time by Meijer in 1936 as an attempt to introduce a very general function so that to include most of the known special functions as a particular case. 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.

1 Definition
- 1.1 The differential equation of the G-function
2 Analytic continuation of the G-function
3 Relationship between G-function and hypergeometric function
4 Elementary properties of the G-function
- 4.1 Multiplication theorem
5 Integrals involving G-function
6 Laplace transform
7 Integral transforms using the G-function
- 7.1 Narain transform
- 7.2 Wimp transform
8 Relationship between the G-function and other elementary functions
9 References

[edit] Definition

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

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

The function is defined under the following hypothesis:

$0 \leq m \leq q$ , $0 \leq n \leq p$ and $p \leq q-1$
$z \neq 0$
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 $\Gamma (1 - a_k + s), k = 1,2,\dots,n$
$a_j - b_k \neq 1,2,3,\dots$ for $j = 1,2,\dots,n$ and $k = 1,2,\dots,m$
if $p = q$ , then the definition makes sense only for $| z | < 1$

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:

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

the L in the integral represents the path to follow while integrating. 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 $\delta = m + n - \frac{1}{2} (p+q)> 0$ , $| \arg z | < \delta \pi$ . Defining:

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

when $| \arg z | = \delta \pi, \delta \geq 0$ :

when $p = q$ , the integral converges if $Re {ν} < - 1$
when $p \neq q$ , expressing $s = σ + i τ$ , where $σ$ and $τ$ are real, the integral converges if, when $\tau \to \infty$ , the following relationship is true:

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

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

| z | < 1

3. L is a loop beginning and ending at $-\infty$ and encircling all poles of $\Gamma (1 - a_k + s), k = 1,2,\dots,n$ , 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:

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

This relationship is valid only when the integral converges among the second path, i.e. when $p < q$ , or when $p = q$ and $| z | < 1$ . The asterisks have a particular meaning. In the product it reminds to ignore the case $b j = b h$ , replacing it with a 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 asterisc reminds to ignore the case $b i = b h$ , replacing it again with a 1.

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

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

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.

[edit] The differential equation of the G-function

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

$\left[ (-1)^{m + n - p} z \prod_{j = 1}^{p} \left( z \frac{d}{dz} - a_j + 1 \right) - \prod_{i = 1}^{q} \left( z \frac{d}{dz} - b_i \right) \right] U(z) = 0$

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

[edit] 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:

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

This property is really important: using it we can transform a G-function with $p > q$ in another one with $p < q$ (or viceversa). In other terms, we can always use the expression of the G-function in terms of the hypergeometric function (that is valid only when the integral converges on the second path) because, if $p > q$ we can convert it to another one with $p < q$ using this property. In the particular case of $p = q$ , the property is still valid provided that $| z | < 1$ .

[edit] Relationship between G-function and hypergeometric function

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

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

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:

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

both relationships are valid if $\;_p F_q (\cdot)$ is defined, i. e. $p \leq q$ or $p = q + 1$ with $0 < | z | < 1$ .

[edit] Elementary properties of the G-function

As is 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, it 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:

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

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 relationships:

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

$G_{p+1,q+1}^{m,n+1} \left( \left. \begin{matrix} a, \mathbf{a_p} \\ \mathbf{b_q}, b \end{matrix} \; \right| \; z \right) = (-1)^r G_{p+1,q+1}^{m+1,n} \left( \left. \begin{matrix} \mathbf{a_p},a \\ b,\mathbf{b_q} \end{matrix} \; \right| \; z \right), q \geq m, a-b = r, r \mbox{ integer or zero}$

about derivatives, there are the following relationships:

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

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

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

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

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:

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

moreover:

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

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

several properties of the hypergeometric function and of other special functions can be deduce by these relationships.

[edit] Multiplication theorem

Provided that $z \neq 0$ , and that m, n, p e q are integer with

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

So the following relationship is valid:

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

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

[edit] Integrals involving G-function

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

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

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

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

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

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

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

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

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

[edit] Laplace transform

Using the previous relationships it is possible to prove that:

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

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

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

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

This is another Laplace transform involving the G-function:

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

[edit] Integral transforms using the G-function

In general, two functions $k (z, y)$ and $h (z, y)$ are called transform kernelsif, 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 $k (z, y) = h (z, y)$ .

[edit] 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.

[edit] 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:

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

[edit] 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:

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

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

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

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

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

$\arcsin x = 2 \sqrt{\pi} G_{2,2}^{1,2} \left( \left. \begin{matrix} 3/2, 3/2 \\ 1, 1/2 \end{matrix} \; \right| \; - x^2 \right) , \qquad |x| < 1$

$\arctan x = 2 G_{2,2}^{1,2} \left( \left. \begin{matrix} 3/2,1 \\ 1,1/2 \end{matrix} \; \right| \; x^2 \right) , \qquad |x| < 1$

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

$J_\alpha (x) = G_{0,4}^{2,0} \left( \left. \begin{matrix} - \\ \frac{\alpha}{4},\frac{\alpha + 2}{4},\frac{-\alpha}{4},\frac{-\alpha + 2}{4} \end{matrix} \; \right| \; \frac{x^4}{256} \right) , \qquad \forall x$

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

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

[edit] References

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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Category: Hypergeometric functions