Symmetric Truncated Freud polynomials

TL;DR

This paper introduces symmetric truncated Freud polynomials orthogonal with respect to a specific weight, exploring their properties, recurrence relations, and electrostatic interpretation of their zeros, contributing to the theory of semiclassical orthogonal polynomials.

Contribution

It defines a new family of symmetric truncated Freud polynomials and analyzes their properties, recurrence relations, and electrostatic zero distribution, highlighting their semiclassical nature.

Findings

Polynomials are of semiclassical class 4.

Explicit recurrence coefficients and moments are derived.

Zeros exhibit electrostatic behavior and dynamics with respect to parameter z.

Abstract

We define the family of symmetric truncated Freud polynomials $P_n(x;z)$, orthogonal with respect to the linear functional $\mathbf{u}$ defined by \begin{equation*} \langle \mathbf{u}, p(x)\rangle = \int_{-z}^z p(x)e^{-x^4}dx, \quad p\in \mathbb{P}, \quad z>0. \end{equation*} The semiclassical character of $P_n (x; z)$ as polynomials of class $4$ is stated. As a consequence, several properties of $P_n (x; z)$ concerning the coefficients $\gamma_n (z)$ in the three-term recurrence relation they satisfy as well as the moments and the Stieltjes function of $\mathbf{u}$ are studied. Ladder operators associated with such a linear functional and the holonomic equation that the polynomials $P_n (x; z)$ satisfy are deduced. Finally, an electrostatic interpretation of the zeros of such polynomials and their dynamics in terms of the parameter $z$ are given.