An analytic approach to the finite R-transform

We revisit Marcus' finite free analogue of Voiculescu $R$-transform from an analytic viewpoint. By relating the finite free Fourier transform to the Laplace transform, we study the finite $R$-transform through logarithmic potentials and Legendre transforms. Under suitable assumptions, we prove that the finite $R$-transform of a polynomial differs from the Voiculescu $R$-transform of its empirical root distribution by $O(N^{-1})$. As an application, we obtain an analytic proof of the convergence of finite free additive convolution to free additive convolution.