Electrostatic Interpretation of Zeros of Orthogonal Polynomials

TL;DR

This paper provides a unified electrostatic interpretation for the zeros of classical orthogonal polynomials and introduces a system of ODEs that rapidly converges to these zeros.

Contribution

It generalizes electrostatic interpretations to a broad class of orthogonal polynomials and presents a new ODE system for efficient zero approximation.

Findings

Zeros satisfy a specific electrostatic equilibrium condition.

Derived ODE system converges exponentially to polynomial zeros.

Includes classical polynomials like Jacobi, Hermite, Legendre, Chebyshev, and Laguerre.

Abstract

We study the differential equation $ - (p(x) y')' + q(x) y' = \lambda y,$ where $p(x)$ is a polynomial of degree at most 2 and $q(x)$ is a polynomial of degree at most 1. This includes the classical Jacobi polynomials, Hermite polynomials, Legendre polynomials, Chebychev polynomials and Laguerre polynomials. We provide a general electrostatic interpretation of zeros of such polynomials: the set of real numbers $\left\{x_1, \dots, x_n\right\}$ satisfies $$ p(x_i) \sum_{k = 1 \atop k \neq i}^{n}{\frac{2}{x_k - x_i}} = q(x_i) - p'(x_i) \qquad \mbox{for all}~ 1\leq i \leq n$$ if and only if they are zeros of a polynomial solving the differential equation. We also derive a system of ODEs depending on $p(x),q(x)$ whose solutions converge to the zeros of the orthogonal polynomial at an exponential rate.