On certain properties of perturbed Freud-type weight: a revisit

TL;DR

This paper investigates properties of perturbed Freud-type orthogonal polynomials, including their differential equations, recurrence relations, moments, and electrostatic zero distribution, expanding understanding of semiclassical orthogonal polynomials.

Contribution

It provides new characterizations and differential equations for perturbed Freud-type polynomials, linking their properties to electrostatic models of zeros.

Findings

Derived nonlinear recurrence relations for perturbed Freud polynomials.

Established differential equations satisfied by these polynomials.

Provided electrostatic interpretation of zeros distribution.

Abstract

In this paper, monic polynomials orthogonal with deformation of the Freud-type weight function are considered. These polynomials fullfill linear differential equation with some polynomial coefficients in their holonomic form. The aim of this work is explore certain characterizing properties of perturbed Freud type polynomials such as nonlinear recursion relations, finite moments, differential-recurrence and differential relations satisfied by the recurrence coefficients as well as the corresponding semiclassical orthogonal polynomials. We note that the obtained differential equation fulfilled by the considered semiclassical polynomials are used to study an electrostatic interpretation for the distribution of zeros based on the original ideas of Stieltjes.