Electrostatic interpretation of zeros of orthogonal polynomials

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: a set of distinct, real numbers $\left \{x_1, \dots , x_n\right \}$ satisfies \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 $p(x),q(x)$ whose solutions converge to the zeros of the orthogonal polynomial at an exponential rate.