Probability distributions with rational free $R$-transform

We study the class ${\mathcal{M}}_{\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}}$ of those probability distributions for which the free $R$-transforms are rational functions. This class is closed under the additive free convolution, additive free powers and under the monotone convolution. We prove a sufficient condition that a rational function is the free $R$-transform of a probability distribution. Several examples are provided, including that of free deconvolution.