On the free Gamma distributions

Abstract

For each positive number $\alpha$ we study the analog ${\nu }_{a}lpha$ 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 $\alpha \downarrow 0$.

31 pages, 4 figures