On the Darboux transformations and sequences of $p$-orthogonal polynomials

TL;DR

This paper explores the relationship between Darboux transformations and sequences of p-orthogonal polynomials, extending the understanding of their recurrence relations and associated functionals in approximation theory.

Contribution

It analyzes how Darboux transformations affect the vector of functionals related to p-orthogonal polynomial sequences, advancing the theoretical framework.

Findings

Darboux transformations modify the functional vectors associated with p-orthogonal polynomials.

The structure of recurrence relations is preserved under certain Darboux transformations.

New insights into the connection between banded matrices and polynomial orthogonality are provided.

Abstract

For a fixed $p \in \mathbb{N}$, sequences of polynomials $\{P_n\}$, $n \in \mathbb{N}$, defined by a $(p+2)$-term recurrence relation are related to several topics in Approximation Theory. A $(p+2)$-banded matrix $J$ determines the coefficients of the recurrence relation of any of such sequences of polynomials. The connection between these polynomials and the concept of orthogonality has been already established through a $p$-dimension vector of functionals. This work goes further in this topic by analyzing the relation between such vectors for the set of sequences $\{P_n^{(j)}\}$, $n \in N$, associated with the Darboux transformations $J^{(j)}$, $j=1, ..., p,$ of a given $(p+2)$-banded matrix $J$.