Generalized Meixner-type free gamma distributions:convolution formulas and potential correspondence

Abstract

We introduce and study a class of generalized Meixner-type free gamma distributions $\mu_{t,\theta,\lambda}$ ($t,\theta>0$ and $\lambda\ge 1$), which includes both the free gamma distributions introduced by Anshelevich and certain scaled free beta prime distributions introduced by Yoshida. We investigate fundamental properties and mixture structures of these distributions. In particular, we consider the Gibbs distribution $\frac{1}{\mathcal{Z}_{t,\theta,\lambda}} \exp\{-V_{t,\theta,\lambda}(x)\}$ associated with a family of potentials $V_{t,\theta,\lambda}$, and show that $\mu_{t,\theta,\lambda}$ maximizes Voiculescu's free entropy with potential $V_{t,\theta,\lambda}$ for parameters $t,\theta>0$ and $1\le \lambda<1+t/\theta$. This result substantially extends the range of classcal-free correspondences obtained the potential function, differing from those arising from the Bercovici-Pata bijection. Moreover, we identify algebraic relations involving noncommutative random variables distributed as free gamma distributions.