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We first introduce and derive some basic properties of a two-parametersfamily of one-sided Levy processes. Their Laplace exponents are given in termsof the Pochhammer symbol. This family includes, in a limit case, the family ofBrownian motion with drifts. Then, we proceed by computing the density of thelaw of the exponential functional associated to some elements of this family(and their dual) and some transformations of these elements. These densitiesare expressed in terms of the Wright hypergeometric functions. By means ofprobabilistic arguments, we derive some interesting properties enjoyed by thesefunctions. On the way we also characterize explicitly the density of thesemi-groups of the family of self-similar continuous state branching processeswith immigration.

Law of the exponential functional of a new family of one-sided Levy  processes via self-similar continuous state branching processes with  immigration and the Wright hypergeometric functions