An electrostatic interpretation of the zeros of sieved ultraspherical polynomials

TL;DR

This paper provides an electrostatic interpretation of the zeros of sieved ultraspherical polynomials, deriving differential equations and analyzing their properties through polynomial transformations.

Contribution

It introduces a unified approach to derive properties of sieved ultraspherical polynomials using polynomial transformations and establishes an electrostatic model for their zeros.

Findings

Zeros correspond to particles in electrostatic equilibrium

Derived differential equations for sieved ultraspherical polynomials

Unified method for analyzing properties of these polynomials

Abstract

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 polynomial transformations. In this work we use this fact to derive in an unified way old and new properties concerning the sieved ultraspherical polynomials of the first and second kind. In particular we derive ordinary differential equations for these polynomials. As an application, we use the differential equation for sieved ultraspherical polynomials of the first kind to deduce that the zeros of these polynomials mark the locations of a set of particles that are in electrostatic equilibrium with respect to a particular external field.