Weighted equilibrium and the flow of derivatives of polynomials

Abstract

Given a sequence of polynomials $Q_n$ of degree $n$ with zeros on $[-1,1]$, we consider the triangular table of derivatives $Q_{n, k}(x)=d^k Q_n(x) /d x^k$. Under the assumption that the sequence $\{Q_n\}$ has a weak* limiting zero distribution (an empirical distribution of zeros) given by the arcsine law, we show that as $n, k \rightarrow \infty$ such that $k / n \rightarrow t \in[0,1)$, the zero-counting measure of the polynomials $Q_{n, k}$ converges to an explicitly given measure $\mu_t$. This measure is the equilibrium measure of $[-1,1]$ of size $1-t$ in an external field given by two mass points of size $t/2$ fixed at $\pm 1$. The main goal of this paper is to provide a direct potential theory proof of this fact.