Stable Densities, Fractional Integrals and the Mittag-Leffler Function

Abstract

This paper combines probability theory and fractional calculus to derive a novel integral representation of the three-parameter Mittag-Leffler function or Prabhakar function, where the three parameters are combinations of four base parameters. The fundamental concept is the Riemann-Liouville fractional integral of the one-sided stable density, conditioned on a scale factor. Integrating with respect to a gamma-distributed scale factor induces a mixture of Riemann-Liouville integrals. A particular combination of four base parameters leads to a representation of the Prabhakar function as a weighted mixture of Riemann-Liouville integrals at different scales. The Prabhakar function constructed in this manner is the Laplace transform of a four-parameter distribution. This general approach gives various known results as special cases (notably, the two-parameter generalised Mittag-Leffler distribution).