Mittag-Leffler function

Remarks

The series has infinite radius of convergence, because $\Gamma(\alpha k + \beta)$ grows faster than any power of $k$ when $\operatorname{Re} \alpha > 0$, so $E_{\alpha, \beta}$ is an entire function.

Example: for $\alpha = \beta = 1$, $\Gamma(k + 1) = k!$, so $E_{1,1}(z) = e^z$. Also $E_2(z) = \sum_k \frac{z^k}{(2k)!} = \cosh \sqrt{z}$, and $E_{1/2}(z) = e^{z^2} \operatorname{erfc}(-z)$.

Non-example: for $\alpha = 0$ and $\beta = 1$ the series is $\sum_k z^k$, which is not entire; it converges only for $\abs{z} < 1$, so the condition $\operatorname{Re} \alpha > 0$ cannot be dropped.

Laplace transform: for $\operatorname{Re} s$ large enough, $$\int_0^\infty e^{-st} \, t^{\beta - 1} E_{\alpha, \beta}(\lambda t^\alpha) \, dt = \frac{s^{\alpha - \beta}}{s^\alpha - \lambda},$$ which follows by integrating the series term by term.