Weighted equilibrium and the flow of derivatives of polynomials
Andrei Martinez-Finkelshtein, Evgenii A. Rakhmanov
TL;DR
This work analyzes how the zeros of the derivative table $Q_{n,k}$ of polynomials $Q_n$ with zeros on $[-1,1]$ move as $k$ grows with $n$, under the arc-sine initial law for $Q_n$ zeros. It proves that, as $k/n\to t\in[0,1)$, the zero-counting measures converge to an explicit absolutely continuous measure $\mu_t$ with density $d\mu_t/dx = \frac{1}{\pi}\frac{\sqrt{1-t^2-x^2}}{1-x^2}$ on $[-\sqrt{1-t^2},\sqrt{1-t^2}]$, and that $\mu_t$ equals the equilibrium measure in the external field $\varphi_t(x)=\frac{t}{2}\log\frac{1}{|x^2-1|}$ with total mass $1-t$. The result is interpreted as the flow of zeros under a potential-theoretic evolution, and is illustrated by the Jacobi polynomial case via the known derivative relation $Q'_n(x)=n P^{(\alpha+1,\beta+1)}_{n-1}(x)$. The authors provide a direct potential-theoretic proof, showing the generic nature of the phenomenon for arc-sine initial distributions and connecting to broader descriptions of the Cauchy transform in related work. These insights clarify the structure of zero distributions under differentiation and offer a rigorous framework for similar flows in orthogonal polynomial families.
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 $μ_t$. 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.