Given freely independent random variables $X_i$ with Marchenko-Pastur measures $\mu_i$, $i\in\{1,\dots,n\}$ how can we find the distribution of the scaled sum of these random variables $\sum_ia_iX_i$. (The measures are the empirical distributions of eigenvalues of symmetric random matrices.)

The answer seems to be free convolution of $\mu_i$, $i\in\{1,\dots,n\}$. To do that we can start with the R-transform. The R-transform of M-P measure $\mu_i$ is $R_i(z)=\frac{a_i}{1-\beta a_i z}$, where $\beta$ is the ratio of the dimensions of the random matrices. Now the R-trans of the sum distribution $R(z)$ is the sum of the R-transforms $R_i(z)$, $R(z)=\sum_i\frac{a_i}{1-\beta a_iz}$.

The question is how can one find the inverse of this $R(z)=\sum_i\frac{a_i}{1-\beta a_iz}$? So finally will have the law of $\sum_ia_iX_i$.

P.S.: Related to Random matrix determinant problem