An electrostatic interpretation of the zeros of sieved ultraspherical polynomials

In an earlier work [Castillo et al., J. Math. Anal. Appl. 455, 1801–1821 (2017)], it was proved that the semiclassical class of orthogonal polynomials is stable under polynomial transformations. In this work, we use this fact to derive in a 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.