Orthogonal polynomials generated by a linear structure relation: Inverse problem

TL;DR

This paper investigates conditions under which two polynomial sequences linked by a linear relation are orthogonal, providing characterizations and examples for their orthogonality properties in the context of inverse problems.

Contribution

It offers necessary and sufficient conditions for the non-degeneracy of the relation and characterizes the orthogonality of the sequences in terms of polynomial coefficients and distributional transformations.

Findings

Conditions for non-degeneracy of the relation when both sequences are orthogonal.

Characterization of orthogonality of the second sequence via polynomial coefficients.

Examples illustrating the theoretical results.

Abstract

Let $(P_n)_n$ and $(Q_n)_n$ be two sequences of monic polynomials linked by a type structure relation such as $$ Q_{n}(x)+r_nQ_{n-1}(x)=P_{n}(x)+s_nP_{n-1}(x)+t_nP_{n-2}(x)\;, $$ where $(r_n)_n$, $(s_n)_n$ and $(t_n)_n$ are sequences of complex numbers. First, we state necessary and sufficient conditions on the parameters such that the above relation becomes non-degenerate when both sequences $(P_n)_n$ and $(Q_n)_n$ are orthogonal with respect to regular moment linear functionals ${\bf u}$ and ${\bf v}$, respectively. Second, assuming that the above relation is non-degenerate and $(P_n)_n$ is an orthogonal sequence, we obtain a characterization for the orthogonality of the sequence $(Q_n)_n$ in terms of the coefficients of the polynomials $\Phi$ and $\Psi$ which appear in the rational transformation (in the distributional sense) $\Phi {\bf u}=\Psi {\bf v}\; .$ Some illustrative examples of the developed theory are presented.