Electrostatic interpretation of zeros of orthogonal polynomials

Abstract

We study the differential equation − ( p ( x ) y ′ ) ′ + q ( x ) y ′ = λ y , - (p(x) y’)’ + q(x) y’ = \lambda y, where p ( x ) p(x) is a polynomial of degree at most 2 and q ( x ) 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: a set of distinct, real numbers { x 1 , … , x n } \left \{x_1, \dots , x_n\right \} satisfies p ( x i ) ∑ k = 1 k ≠ i n 2 x k − x i = q ( x i ) − p ′ ( x i ) f o r a l l 1 ≤ i ≤ n \begin{equation*} p(x_i) \sum _{k = 1 \atop k \neq i}^{n}{\frac {2}{x_k - x_i}} = q(x_i) - p’(x_i) \qquad \mathrm {for all}~ 1\leq i \leq n \end{equation*} 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 ) p(x),q(x) whose solutions converge to the zeros of the orthogonal polynomial at an exponential rate.