Some analytical and numerical properties of the Mittag- Leffler functions

Some analytical properties of the Mittag-Leffler functions, $e_{\alpha}(t) \equiv E_{\alpha}(-t^{\alpha})$, are established on some $t$-intervals. These are lower and upper bounds obtained in terms of simple rational functions (which are related to the Pad\'e approximants of $e_{\alpha}(t)$), for $t > 0$ and $0 < \alpha < 1$. A new method, to compute such functions, solving numerically a Caputo-type fractional differential equation satisfied by them, is developed. This approach consists in an {\em adaptive} predictor-corrector method, based on the K.~Diethelm's predictor-corrector algorithm, and is shown to outperform the current methods implemented by MATLAB$^{\textregistered}$ and by {\em Mathematica} when $t$ is real and even possibly large.

Some analytical properties of the Mittag-Leffler functions, eα(t) ≡ Eα(-tα), are established on some t-intervals. These are lower and upper bounds obtained in terms of simple rational functions (which are related to the Pad� approximants of eα(t)), for t > 0 and 0 < α < 1. A new method, to compute such functions, solving numerically a Caputo-type fractional differential equation satisfied by them, is developed. This approach consists in an adaptive predictor-corrector method, based on the K. Diethelm's predictor-corrector algorithm, and is shown to outperform the current methods implemented by MATLAB� and by Mathematica when t is real and even possibly large.