Orthogonal polynomials associated with an inverse quadratic spectral transform

TL;DR

This paper characterizes when a perturbed sequence of orthogonal polynomials remains orthogonal, using quadratic polynomial relations and explores implications for Jacobi matrices, with practical examples.

Contribution

It provides a new characterization of orthogonality for perturbed polynomial sequences via quadratic polynomial relations and their associated linear functionals.

Findings

New criteria for orthogonality of perturbed polynomial sequences

Explicit cases where parameters are easier to compute

Connections between perturbations and Jacobi matrices

Abstract

Let $\{P_n \}_{n\ge0}$ be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional $u$ and $\{Q_n \}_{n\ge0}$ a sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n P_{n-1}(x)+t_n P_{n-2}(x),\quad n\ge1,$$ with $t_n \not= 0$ for $n\ge2$. We obtain a new characterization of the orthogonality of the sequence $\{Q_n \}_{n\ge0}$ with respect to a linear functional $v$, in terms of the coefficients of a quadratic polynomial $h$ such that $h(x)v= u$. We also study some cases in which the parameters $s_n$ and $t_n$ can be computed more easily, and give several examples. Finally, the interpretation of such a perturbation in terms of the Jacobi matrices associated with $\{P_n \}_{n\ge0}$ and $\{Q_n \}_{n\ge0}$ is presented.