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Some Analytical and Numerical Properties of the Mittag-Leffler Functions

Abstract 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 Padé 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