On the free Gamma distributions

Abstract

For each positive number $\alpha$ we study the analog $\nu_alpha$ in free probability of the classical Gamma distribution with parameter $\alpha$. We prove that $\nu_\alpha$ is absolutely continuous and establish the main properties of the density, including analyticity and unimodality. We study further the asymptotic behavior of $\nu_\alpha$ as $\alpha\downarrow0$.