Electrostatic partners and zeros of orthogonal and multiple orthogonal polynomials

TL;DR

This paper develops a general framework linking orthogonal and multiple orthogonal polynomial zeros to electrostatic models via second-order differential equations, extending known results and exploring asymptotic behaviors.

Contribution

It introduces a universal construction of second-order ODEs for polynomial zeros and their electrostatic interpretation, generalizing previous results for various polynomial systems.

Findings

Derived electrostatic models for zeros of orthogonal and multiple orthogonal polynomials.

Extended known results to Angelesco, Nikishin, and generalized Nikishin systems.

Analyzed the asymptotic transition to vector equilibrium problems.

Abstract

For a given polynomial $P$ with simple zeros, and a given semiclassical weight $w$, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of $P$. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of $P$. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of $P$ with respect to $w$ allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite--Pad\'e or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems. We also discuss the discrete-to-continuous transition of these models in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.