On the Mittag-Leffler distributions

We first prove that the Mittag-Leffler distributions belong to the class of distributions with complete monotone derivative. Then we investigate the fundamental properties of the Mittag-Leffler distributions and of their extensions, including the tail behavior of distribution, the explicit expressions for moments of all orders and for the density functions. The latter has been used to correct some inverse Laplace transforms given in the literature. As a by-product, the moments of negative (integral) orders are used to characterize the positive stable distributions with exponent alpha is an element of[1/3, 1]. (C) 1998 Elsevier Science B.V. All rights reserved.