Fox–Wright Function

In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function _''p''''F''_''q''(''z'') based on ideas of and :

_p\Psi_q \left begin
( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\
( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end
; z \right=
\sum_^\infty \frac \, \frac .

Upon changing the normalisation

_p\Psi^*_q \left begin
( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\
( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end
; z \right=
\frac
\sum_^\infty \frac \, \frac

it becomes _''p''''F''_''q''(''z'') for ''A''_1...''p'' = B_1...''q'' = 1.

The Fox–Wright function is a special case of the Fox H-function :

_p\Psi_q \left begin
( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\
( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end
; z \right=
H^_ \left \begin
( 1-a_1 , A_1 ) & ( 1-a_2 , A_2 ) & \ldots & ( 1-a_p , A_p ) \\
(0,1) & (1- b_1 , B_1 ) & ( 1-b_2 , B_2 ) & \ldots & ( 1-b_q , B_q ) \end \right. \right

A special case of Fox-Wright function appears as a part of the normalizing constant of the Modified half-normal distribution with the pdf on $(0, \infty)$ is given as $f(x)= \frac$, where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox-Wright Psi function.

Wright function

The entire function $W_(z)$ is often called the Wright function. It is the special case of _0\Psi_1 \left ..\right/math> of the Fox-Wright function. Its series representation is

$W_(z) = \sum_^\infty \frac, \lambda > -1.$

This function is used extensively in fractional calculus and the stable count distribution.

Three properties were stated in Theorem 1 of Wright (1933) and 18.1(30-32) of Erdelyi, Bateman Project, Vol 3 (1955) (p.212)

$\begin \lambda z W_(z) & = & W_(z) + (1-\mu) W_(z) & (a) \\ W_(z) & = & W_(z) & (b) \\ \lambda z W_(z) & = & W_(z) + (1-\mu) W_(z) & (c) \end$

Equation (a) is a recurrence formula. (b) and (c) provide two paths to reduce a derivative. And (c) can be derived from (a) and (b).

A special case of (a) is $\lambda = -\alpha, \mu = 1$. Replacing $z$ with $-z$, we have
$-\alpha z W_(-z) = W_(-z)$

Two notations, $M_(z)$ and $F_(z)$, were used extensively in the literatures:

\begin
M_(z) & = W_(-z), \\ ex\implies
F_(z) & = W_(-z) = \alpha z M_(z).
\end

M-Wright function

$M_(z)$ is known as the M-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes, generally referred to as time-fractional diffusion processes.

Its properties were surveyed in Mainardi et al (2010).
Through the stable count distribution, $\alpha$ is connected to Lévy's stability index $(0 < \alpha \leq 1)$.

Its asymptotic expansion of $M_(z)$ for $\alpha > 0$ is
$M_\left ( \frac \right ) = A(\alpha) \, r^ \, e^, \,\, r\rightarrow \infty,$
where
$A(\alpha) = \frac,$
$B(\alpha) = \frac.$

See also

* Hypergeometric function
* Generalized hypergeometric function
* Modified half-normal distribution with the pdf on $(0, \infty)$ is given as $f(x)= \frac$, where $\Psi(\alpha,z)=_1\Psi_1\left(\begin\left(\alpha,\frac\right)\\(1,0)\end;z \right)$ denotes the Fox-Wright Psi function.

References

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External links

hypergeom
on GitLab

{{DEFAULTSORT:Fox-Wright function
Factorial and binomial topics
Hypergeometric functions
Series expansions