Differential orthogonality: Laguerre and Hermite cases with applications

TL;DR

This paper investigates the properties of orthogonal polynomials related to Laguerre and Hermite operators, exploring their algebraic, analytic, asymptotic behaviors, and modeling their zeros with fluid dynamics concepts.

Contribution

It introduces a unified study of orthogonal polynomials associated with differential operators, revealing new algebraic and asymptotic properties and linking zeros to fluid dynamics models.

Findings

Characterization of orthogonal polynomials for Laguerre and Hermite cases.

Asymptotic behavior of polynomial zeros analyzed.

Fluid dynamics model for zeros of these polynomials developed.

Abstract

Let $\mu$ be a finite positive Borel measure supported on R, $\LL[f] =xf''+(\alpha+1-x)f'$ with $\alpha>-1$, or $\LL[f] =\frac{1}{2}f''-xf'$, and $m$ a natural number. We study algebraic, analytic and asymptotic properties of the sequence of monic polynomials $\{Q_n\}_{n>m}$ orthogonal with respect to the operator $\LL$. We also provide a fluid dynamics model for the zeros of these polynomials.