An electrostatics model for zeros of general orthogonal polynomials

Abstract

We prove that the zeros of general orthogonal polynomials, subject to certain integrability conditions on their weight functions determine the equilibrium position of movable n unit charges in an external field determined by the weight function. We compute the total energy of the system in terms of the recursion coefficients of the orthonormal polynomials and study its limiting behavior as the number of particles tends to infinity in the case of Freud exponential weights.

Lamé differential equations and electrostatics

“…We refer to Chapter 2.9 of Marden's monograph [9] for more detailed information about Lamé's equation and to [19,17] for the connection with electrostatics of zeros of orthogonal polynomials. In very recent papers Grünbaum [3] gave an electrostatic interpretation of the zeros of Koornwinder-Krall polynomials and Ismail [5] showed that the zeros of a general class of orthogonal polynomials are points of unique equilibrium in an electrostatic field and calculated the energy of the field. Grünbaum [4] established a result on electrostatics of zeros of the polynomials obtained from those of Jacobi by repeated applications of the Darboux transformation.…”

Structure relations for orthogonal polynomials

“…The differential equations (2-3) were used to determine the equilibrium position of the particles in a Coulomb gas model; see [Ismail 2000]. …”

Inequalities for Extreme Zeros of Some Classical Orthogonal and q-orthogonal Polynomials

“…[46,Chapter 6]) and generalised by Ismail (cf. [27,28]) provide further reasons for interest in the properties of the zeros.…”