  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).
