Fox–Wright function

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 $E_{\alpha, \beta}(z)$; 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.

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 $z \neq 0$.