On the Darboux transformations and sequences of p-orthogonal polynomials

For a fixed p∈N, sequences of polynomials {Pn}, n∈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 already been established through a p-dimension vector of functionals. This work goes further on this topic by analyzing the relation between such vectors for the set of sequences {Pn(j)},n ∈ N, associated with the Darboux transformations J(j), j=1,…,p, of a given (p+2)-banded matrix J. This is synthesized in Theorem 1, where, under certain conditions, these relationships are established. Besides, some relationships between the sequences of polynomials {Pn(j)} are determined in Theorem 2, which will be of interest for future research on p-orthogonal polynomials. We also provide an example to illustrate the effect of the Darboux transformations of a Hessenberg banded matrix, showing the sequences of p-orthogonal polynomials and the corresponding vectors of functionals. For the sake of clarity, in this example we have considered the case p=2, since the procedure is similar for p > 2.