arXiv subjects

D. Barrios Rolanía

Publications and source records attributed to D. Barrios Rolanía.

8 recordsLinked to original sources

arXiv2026

Finite-term recurrences in a generalized Bochner--Krall family

We classify the differential operators $T=z^j\partial_z^j+z^m\partial_z^\ell$, where $0\le m<\ell$ and $1\le j<\ell$, whose monic eigenpolynomials satisfy a finite-term recurrence relation. Writing $k=\ell-m$, such a recurrence exists if and only if $j=1$ and $k\mid\ell$. In that case we determine all recurrence coefficients in closed form and prove that the associated difference operator has order $\ell$. We also give an explicit factorization showing that every admissible operator is of Type~(2) in Conjecture~1.10 of Horozov--Shapiro--Tater; the equality of the differential and difference orders is the conclusion predicted by their Conjecture~1.9.

math-ph↗

arXiv2024

New tools for the study of Bochner differential operators

A sequence $\{δ_n^{(k)}\}$ associated to a Bochner differential operator is introduced as an effective tool to study this kind of operators. Some properties of this sequence are proven and used to deduce that a particular operator leads to solutions of a bispectral problem. In addition, the inverse problem is studied; that is, given a sequence $\{λ_n\}$ of complex numbers and a sequence $\{P_n\}$ of polynomials with complex coefficients, $°{P_n}=n$, we find a necessary and sufficient condition for the existence of a Bochner differential operator that has those sequences as eigenvalues and eigenpolynomials, respectively. The mentioned condition also depends on $\{δ_n^{(k)}\}$.

math.FA↗

arXiv2023

On polynomial solutions of certain finite order ordinary differential equations

Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial eigenfunctions are explicitly obtained, as well as the corresponding eigenvalues. Also, some linear transformations are applied to sequences of eigenfunctions and a necessary condition for this to be a sequence of eigenfunctions of a new differential operator is obtained. These results are applied to the particular case of classical Hermite polynomials.

math.FA↗

arXiv2021

The bispectral problem and polynomial solutions

Some properties and relations satisfied by the polynomial solutions of the bispectral problem are studied. Given a differential operator, under certain restrictions its polynomial eigenfunctions are explicitly obtained, as well as the corresponding eigenvalues. Also the inverse problem is approached, giving conditions for the existence of a differential operator from its eigenpolynomials and eigenvalues.

math.SP↗

arXiv2021

A note on banded linear systems

In [2] a new factorization for infinite Hessenberg banded matrices was introduced. In this note we prove that this kind of factorization can also be used for finite matrices. In addition, a new method for solving banded linear systems is provided.

math.NA↗

arXiv2019

On the Darboux transformations and sequences of $p$-orthogonal polynomials

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 $J$.

math.FA↗

arXiv2019

Geronimus transformations for sequences of $d$-orthogonal polynomials

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,\ldots, d,$ are analyzed and some relationships between these new sequences of polynomials are given. Also the associated Hessenberg matrices and their transformed are studied.

math.FA↗

arXiv2016

High order recurrence relation, Hermite-Padé approximation, and Nikishin systems

The study of sequences of polynomials satisfying high order recurrence relations is connected with the asymptotic behavior of multiple orthogonal polynomials, the convergence properties of type II Hermite-Padé approximation, and eigenvalue distribution of banded Toeplitz matrices. We present some results for the case of recurrences with constant coefficients which match what is known for the Chebyshev polynomials of the first kind. In particular, under appropriate assumptions, we show that the sequence of polynomials satisfies multiple orthogonality relations with respect to a Nikishin system of measures.

math.CV↗