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 Qn, k(x)=dᵏ Qₙ(x) /d xᵏ. Under the only assumption that the sequence ₙ\ has a weak* limiting zero distribution (an empirical distribution of zeros) represented by a unit measure μ₀ 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 Qn, k converges in a neighborhood of infinity to the Cauchy transform of a measure μₜ. The family of measures μₜ, 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 μ₀, or a nonlocal diffusion equation governing the density of μₜ on R. The main goal of this paper is to provide a streamlined and elementary proof of all these facts.