Symmetrization process and truncated orthogonal polynomials

TL;DR

This paper introduces truncated Laguerre polynomials and their symmetrized forms, analyzing their properties, asymptotics, and zero distributions, with connections to Painlevé equations and electrostatic models.

Contribution

It defines new truncated Laguerre polynomials and explores their properties, asymptotics, and zero behavior, linking them to Painlevé equations and electrostatic interpretations.

Findings

Properties of truncated Laguerre polynomials and their symmetrized counterparts are characterized.

Asymptotic expansions of recurrence coefficients are derived.

Zeros of the polynomials are interpreted via electrostatic models and their dynamics analyzed.

Abstract

We define the family of truncated Laguerre polynomials $P_n(x;z)$, orthogonal with respect to the linear functional $\ell$ defined by $$\langle{\ell,p\rangle}=\int_{0}^zp(x)x^\alpha e^{-x}dx,\qquad\alpha>-1.$$ The connection between $P_n(x;z)$ and the polynomials $S_n(x;z)$ (obtained through the symmetrization process) constitutes a key element in our analysis. As a consequence, several properties of the polynomials $P_n(x;z)$ and $S_n(x;z)$ are studied taking into account the relation between the parameters of the three-term recurrence relations that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlev\'e and Painlev\'e equations associated with such coefficients appear in a natural way. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameter $z$ are given.