We consider here the recently proposed closed form formula in terms of the
Meijer G-functions for the probability density functions gα(x) of
one-sided L\'evy stable distributions with rational index α=l/k, with
0<α<1. Since one-sided L\'evy and Mittag-Leffler distributions are known
to be related, this formula could also be useful for calculating the
probability density functions ρα(x) of the latter. We show, however,
that the formula is computationally inviable for fractions with large
denominators, being unpractical even for some modest values of l and k. We
present a fast and accurate numerical scheme, based on an early integral
representation due to Mikusinski, for the evaluation of gα(x) and
ρα(x), their cumulative distribution function and their derivatives
for any real index α∈(0,1). As an application, we explore some
properties of these probability density functions. In particular, we determine
the location and value of their maxima as functions of the index α. We
show that α≈0.567 and α≈0.605 correspond,
respectively, to the one-sided L\'evy and Mittag-Leffler distributions with
shortest maxima. We close by discussing how our results can elucidate some
recently described dynamical behavior of intermittent systems.Comment: 6 pages, 5 figures. New references added, final version to appear in
PRE. Numerical code available at http://vigo.ime.unicamp.br/dist