Title: On the free Gamma distributions

Authors: Steen Thorbjornsen and Uffe Haagerup

Issue: Volume 63 (2014), Issue 4, 1159-1194


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 $\alpha\downarrow0$.