On the Darboux transformations and sequences of p-orthogonal polynomials

• Published in: Applied mathematics and computation 2020-10, Vol.382, p.125337, Article 125337
• Main Authors: Barrios Rolanía, D., García-Ardila, J.C., Manrique, D.
• Format: Article
• Language: English
• Subjects: Darboux transformations; Hessenberg banded matrices; p-orthogonal polynomials
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2020.125337

•The Darboux factorization for Hessenberg banded matrices is used.•The effect of this factorization on the sequence of polynomials associated to each matrix is analyzed.•The concept of Darboux transformation for vectors of functionals is introduced and analyzed.•An example is introduced to illustrate the above concepts and the related results. 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.