Regularity and Convergence Properties of Finite Free Convolutions

Finite free convolutions, [Formula: see text] and [Formula: see text], are binary operations on polynomials of degree [Formula: see text] that are central to finite free probability, a developing field at the intersection of free probability and the geometry of polynomials. Motivated by established regularities in free probability, this paper investigates analogous regularities for finite free convolutions. Key findings include triangle inequalities for these convolutions and necessary and sufficient conditions regarding atoms of probability measures. Applications of these results include proving the weak convergence of [Formula: see text] and [Formula: see text] to their infinite counterparts [Formula: see text] and [Formula: see text] as [Formula: see text], without compactness assumptions. Furthermore, this weak convergence is strengthened to convergence in Kolmogorov distance.