Geronimus transformations for sequences of d-orthogonal polynomials

Abstract

In this paper an extension of the concept of Geronimus transformation for sequences of d-orthogonal polynomials {P n } is introduced. The transformed sequences {P (k) n }, for k = 1, . . . , d, are analyzed and some relationships between these new sequences of polynomials are given. Also the associated Hessenberg matrices and their transformed are studied.

On the Darboux transformations and sequences of p-orthogonal polynomials

“…In the present work we are concerned about finding relations between the vectors of p-orthogonality associated with the Darboux transformations of such matrices (5). Thereby this paper complements the analysis that has been done in [4] for the Geronimus transformations. We include here the following summary, with the more relevant concepts, for an independent reading.…”

“…The use of discrete Darboux transformations was proposed in [11,12] with the focus in the application to the Toda lattices. In [1][2][3][4][5][6] this study was extended to (p + 2)banded matrices as (5). In the present work we are concerned about finding relations between the vectors of p-orthogonality associated with the Darboux transformations of such matrices (5).…”