Flow of the zeros of polynomials under iterated differentiation

Given a sequence of polynomials Qₙ of degree n, we consider the triangular table of derivatives Q₍, ₊ (x) =dᵏ Qₙ (x) /d xᵏ. Under the only assumption that the sequence \Qₙ\ has a weak* limiting zero distribution (an empirical distribution of zeros) represented by a unit measure ₀ with compact support in the complex plane, we show that as n, k such that k / n t (0, 1), the Cauchy transform of the zero-counting measure of the polynomials Q₍, ₊ converges in a neighborhood of infinity to the Cauchy transform of a measure ₜ. The family of measures ₜ, t (0, 1), whose dependence on the parameter t can be interpreted as a flow of the zeros under iterated differentiation, has several interesting connections with the inviscid Burgers equation, the fractional free convolution of ₀, or a nonlocal diffusion equation governing the density of ₜ on R. The main goal of this paper is to provide a streamlined and elementary proof of all these facts.