On the Darboux transformations and sequences of $p$-orthogonal polynomials (1910.03039v1)

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 $J$.