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 Lévy stable distributions with rational index α=l/k, with 0<α<1. Since one-sided Lévy 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 g α(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 0.605 correspond, respectively, to the one-sided Lévy and Mittag-Leffler distributions with shortest maxima. We close by discussing how our results can elucidate some recently described dynamical behavior of intermittent systems. © 2011 American Physical Society.842 Albeverio, S., Casati, G., Merlini, D., (1986) Stochastic Processes in Classical and Quantum Systems, , eds., Springer, BerlinShlesinger, M.F., Zaslavsky, G.M., Frisch, U., (1995) Lévy Flights and Related Topics in Physics, , eds. Springer, BerlinBarndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I., (2001) Levy Processes: Theory and Applications, , eds., Birkhäuser, BostonBardou, F., Bouchaud, J.-P., Aspect, A., Cohen-Tannoudji, C., (2002) Lévy Statistics and Laser Cooling, , Cambridge University Press, Cambridge, EnglandPenson, K.A., Górska, K., (2010) Phys. Rev. Lett., 105, p. 210604. , PRLTAO 0031-9007 10.1103/PhysRevLett.105.210604Barkai, E., (2001) Phys. Rev. e, 63, p. 046118. , PLEEE8 1539-3755 10.1103/PhysRevE.63.046118http://library.wolfram.com/infocenter/MathSource/4377/http://math.bu.edu/people/mveillet/html/alphastablepub.htmlPrudnikov, A.P., Brychkov, A., Marichev, O.I., (1998) Integrals and Series, , Namely, formula 2.2.1.19 in, Yu., Gordon and Breach, AmsterdamOlver, F.W.J., (2010) NIST Handbook of Mathematical Functions, , Cambridge University Press, New YorkFeller, W., (1971) An Introduction to Probability Theory and Its Applications Vol. II, , Wiley, New YorkMikusinski, J., (1959) Stud. Math., 18, p. 191West, B.J., Grigolini, P., Metzler, R., Nonnenmacher, T.F., (1997) Phys. Rev. e, 55, p. 99. , PLEEE8 1539-3755 10.1103/PhysRevE.55.99Mainardi, F., Paradisi, P., Gorenflo, R., preprint arXiv: 0704.0320Nolan, J.P., (1997) Commun. Statist.-Stochastic Models, 13, p. 759. , PRLTAO 1532-6349 10.1080/15326349708807450http://academic2.american.edu/~jpnolan/stable/stable.htmlhttp://www.netlib.org/slatec/guidehttp://vigo.ime.unicamp.br/distrKorabel, N., Barkai, E., (2009) Phys. Rev. Lett., 102, p. 050601. , PRLTAO 0031-9007 10.1103/PhysRevLett.102.050601Korabel, N., Barkai, E., (2010) Phys. Rev. e, 82, p. 016209. , PLEEE8 1539-3755 10.1103/PhysRevE.82.016209Aaronson, J., (1997) An Introduction to Infinite Ergodic Theory, , American Mathematical Society, Providence, RIAkimoto, T., Aizawa, Y., (2007) J. Korean Phys. Soc., 50, p. 254. , KPSJAS 0374-4884 10.3938/jkps.50.254