Electrostatic models for zeros of polynomials: Old, new, and some open problems

  title={Electrostatic models for zeros of polynomials: Old, new, and some open problems},
  author={Francisco Marcell{\'a}n and Andrei Mart{\'i}nez-Finkelshtein and Pedro Mart{\'i}nez-Gonz{\'a}lez},
  journal={Journal of Computational and Applied Mathematics},

Figures from this paper

A Unified Approach to Computing the Zeros of Classical Orthogonal Polynomials

The authors present a unified method for calculating the zeros of the classical orthogonal polynomials based upon the electrostatic interpretation and its connection to the energy minimization

Approximate Closed-Form Formulas for the Zeros of the Bessel Polynomials

These zeros are first computed numerically through an implementation of the electrostatic interpretation formulas and then a fit to the real and imaginary parts as functions of , and it is shown that the resulting complex number is -convergent to for fixed .

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

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

An electrostatic interpretation of the zeros of sieved ultraspherical polynomials

In a companion paper [On semiclassical orthogonal polynomials via polynomial mappings, J. Math. Anal. Appl. (2017)] we proved that the semiclassical class of orthogonal polynomials is stable under

Stable Equilibria for the Roots of the Symmetric Continuous Hahn and Wilson Polynomials

We show that the gradient flows associated with a recently found family of Morse functions converge exponentially to the roots of the symmetric continuous Hahn polynomials. By symmetry reduction the

Solvable dynamical systems and isospectral matrices defined in terms of the zeros of orthogonal or otherwise special polynomials.

Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed.

The Diagonal General Case of the Laguerre-Sobolev Type Orthogonal Polynomials

We consider the family of polynomials orthogonal with respect to the Sobolev type inner product corresponding to the diagonal general case of the Laguerre-Sobolev type orthogonal polynomials. We

Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials

We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials – polynomial solutions of second order differential equations with complex polynomial coefficients. In the case when all



An electrostatics model for zeros of general orthogonal polynomials

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

An electrostatic interpretation of the zeros of the Freud-type orthogonal polynomials.

Polynomials orthogonal with respect to a perturbation of the Freud weight function by the addition of a mass point at zero are considered. These polynomials, called Freud-type orthogonal polynomials,

Asymptotic Properties of Heine-Stieltjes and Van Vleck Polynomials

We study the asymptotic behavior of the zeros of polynomial solutions of a class of generalized Lam? differential equations, when their coefficients satisfy certain asymptotic conditions. The limit

Ladder operators and differential equations for orthogonal polynomials

Under some integrability conditions we derive raising and lowering differential recurrence relations for polynomials orthogonal with respect to a weight function supported in the real line. We also

Zeros of Stieltjes and Van Vleck polynomials and applications


A class of hypergeometric-type differential equations is considered. It is shown that its polynomial solutions yn exhibit an orthogonality with respect to a "varying measure" (a sequence of measures)

A minimum energy problem and Dirichlet spaces

We analyze a minimum energy problem for a discrete electrostatic model in the complex plane and discuss some applications. A natural characteristic distinguishing the state of minimum energy from

On the limit distributions of the zeros of Jonquiegre polynomials and generalized classical orthogonal polynomials

Abstract Jonquiere polynomials Jk are defined by the rational function ∑∞0nkzn = Jk(z)/(1 − z)k+1, k ∈ N 0. For a general class of polynomials including Jk, the limit distribution of its zeros is

Asymptotic Statistics of Zeroes for the Lamé Ensemble

Abstract: The Lamé polynomials naturally arise when separating variables in Laplace's equation in elliptic coordinates. The products of these polynomials form a class of spherical harmonics, which