Asymptotic density of zeros of half range generalized Hermite polynomials

TL;DR

This paper analyzes the asymptotic distribution of zeros of generalized Hermite polynomials under specific weight conditions, providing explicit density formulas and linking zero distribution to the system's energy in an external field.

Contribution

It explicitly derives the global zero density of generalized Hermite polynomials and connects zero distribution with the asymptotic energy of a charge system in an external field.

Findings

Explicit global density of zeros derived

Asymptotic energy of charge system computed

Zeros related to equilibrium configurations

Abstract

We investigate the global density of zeros of generalized Hermite orthogonal polynomials, subject to certain truncated conditions on its weight. We shall given explicitly the global density of zeros under some asymptotic conditions on the weight. Moreover we compute the asymptotic of the total energy of the equilibrium position of the system of $n$ movable unit charges in an external field determined by the weight of the generalized Hermite polynomials. We will see that for finite $n$ the energy is in direct relationship with the zeros of the orthogonal polynomials.