On orthogonal polynomials with respect to a class of differential operators

TL;DR

This paper investigates orthogonal polynomials related to a class of differential operators, analyzing their uniqueness, zero distribution, and conditions for infinite orthogonal sequences, with special focus on first-order cases.

Contribution

It provides a classification of measures for which orthogonal polynomials exist for these operators and studies zero locations and asymptotics, including finite polynomial sequences.

Findings

Classification of measures for infinite orthogonal polynomial sequences

Location of zeros for first-order differential operator polynomials

Asymptotic behavior of these orthogonal polynomials

Abstract

We consider orthogonal polynomials with respect to a linear differential operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}\rho_{k}(z)\frac{d^k}{dz^k}, $$ where $\{\rho_k\}_{k=0}^{M}$ are complex polynomials such that $deg[\rho_k]\leq k, 0\leq k \leq M$, with equality for at least one index. We analyze the uniqueness and zero location of these polynomials. An interesting phenomenon occurring in this kind of orthogonality is the existence of operators for which the associated sequence of orthogonal polynomials reduces to a finite set. For a given operator, we find a classification of the measures for which it is possible to guarantee the existence of an infinite sequence of orthogonal polynomials, in terms of a linear system of difference equations with varying coefficients. Also, for the case of a first-order differential operator, we locate the zeros and establish the strong asymptotic behavior of these polynomials.