We first introduce and derive some basic properties of a two-parameters ( α , γ ) family of one-sided Lévy processes, with 1 < α < 2 and γ > − α . Their Laplace exponents are given in terms of the Pochhammer symbol as follows ψ ( γ ) ( λ ) = c ( ( λ + γ ) α − ( γ ) α ) , λ ⩾ 0 , where c is a positive constant, ( λ ) α = Γ ( λ + α ) Γ ( λ ) stands for the Pochhammer symbol and Γ for the Gamma function. These are a generalization of the Brownian motion, since in the limit case α → 2 , we end up to the Laplace exponent of a Brownian motion with drift γ + 1 2 . Then, we proceed by computing the density of the law of the exponential functional associated to some elements of this family (and their dual) and some transformations of these elements. More precisely, we shall consider the Lévy processes which admit the following Laplace exponent, for any δ > α − 1 α , ψ ( 0 , δ ) ( λ ) = ψ ( 0 ) ( λ ) − α δ λ + α − 1 ψ ( 0 ) ( λ ) , λ ⩾ 0 . These densities are expressed in terms of the Wright hypergeometric functions. By means of probabilistic arguments, we derive some interesting properties enjoyed by these functions. On the way, we also characterize explicitly the semi-group of the family of self-similar continuous state branching processes with immigration.