---
tag: WYMC18W643
kind: definition
name: Fox–Wright function
msc: ["33C20", "33E12"]
defines: XDY1GDM2JW
uses: [0H5SYBSED9, 0MZB06ST2B]
sources:
  - url: https://en.wikipedia.org/wiki/Fox%E2%80%93Wright_function
---

## Definition

Let $p, q \ge 0$ be integers.
Let $a_1, \ldots, a_p, b_1, \ldots, b_q \in \C$ and let $A_1, \ldots, A_p, B_1, \ldots, B_q > 0$ be real.
Assume that $a_i + A_i n$ is never a pole of the [[gamma function|0H5SYBSED9]] $\Gamma$, for $n = 0, 1, 2, \ldots$, and use the convention $1/\Gamma(-m) = 0$ for $m = 0, 1, 2, \ldots$.
The *Fox–Wright function* is the [[power series|0MZB06ST2B]]
$${}_p\Psi_q\left[\begin{matrix} (a_1, A_1), \ldots, (a_p, A_p) \\ (b_1, B_1), \ldots, (b_q, B_q) \end{matrix}; z\right] = \sum_{n=0}^{\infty} \frac{\Gamma(a_1 + A_1 n) \cdots \Gamma(a_p + A_p n)}{\Gamma(b_1 + B_1 n) \cdots \Gamma(b_q + B_q n)} \, \frac{z^n}{n!}.$$

## Remarks

Let $\Delta = 1 + \sum_j B_j - \sum_i A_i$.
If $\Delta > 0$ the series converges for every $z \in \C$.
If $\Delta = 0$ it converges for $\abs{z} < \prod_i A_i^{-A_i} \prod_j B_j^{B_j}$.

Example: with $p = q = 1$, $(a_1, A_1) = (1, 1)$ and $(b_1, B_1) = (\beta, \alpha)$, the coefficient is $\frac{\Gamma(1 + n)}{\Gamma(\beta + \alpha n)} \frac{1}{n!} = \frac{1}{\Gamma(\alpha n + \beta)}$, so the series is the [[Mittag-Leffler function|12G9M694YG]] $E_{\alpha, \beta}(z)$; here $\Delta = \alpha > 0$.

When all $A_i$ and $B_j$ equal $1$,
$${}_p\Psi_q = \frac{\Gamma(a_1) \cdots \Gamma(a_p)}{\Gamma(b_1) \cdots \Gamma(b_q)} \, {}_pF_q(a_1, \ldots, a_p; b_1, \ldots, b_q; z),$$
a multiple of the [[generalized hypergeometric function|VDBAW9PN88]].

Non-example: with $p = 1$, $q = 0$ and $A_1 = 2$, one has $\Delta = -1 < 0$, and the series $\sum \Gamma(1 + 2n) z^n / n!$ diverges for every $z \neq 0$.