Mittag-Leffler function

Summary

The Mittag-Leffler function $E_{\alpha, \beta}(z) = \sum_{k \ge 0} \frac{z^k}{\Gamma(\alpha k + \beta)}$ generalizes the exponential function, which is the case $\alpha = \beta = 1$. It is an entire function for $\operatorname{Re} \alpha > 0$. It plays the role in fractional calculus that the exponential plays in ordinary calculus: it solves the basic linear fractional differential equations.