Mittag-Leffler function

The Mittag-Leffler function $E_{\alpha \beta }$ is a complex function which depends on two complex parameters $\alpha$ and $\beta$. It may be defined by the following series when the real part of $\alpha$ is strictly positive:

$$E_{\alpha \beta }(z)=\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }(\alpha k+\beta )}$$

In this case, the series converges for all values of the argument $z$, so the Mittag-Leffler function is an entire function.