On the Self-Decomposability of Euler's Gamma Function

Abstract

Let Γ be Euler's Gamma function. We prove that, for all α ≠ 0, β > 0, γ > 0, δ > 0, the function (Γ(γ + iαz)/Γ(γ)βiαz)δ, z ∈ R1, is a self-decomposable characteristic function from the Thorin class \(\mathcal{T}_e \) and derive its explicit canonical form. Similarly to [1], we also describe several classes of Levy-type stochastic processes related to Γ.