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 {P^(j)_n}, 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 {P^(j)_n} 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.
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 {P^(j)_n}, 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 {P^(j)_n} 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. Read More
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