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Juan Carlos García-Ardila

Publications and source records attributed to Juan Carlos García-Ardila.

5 recordsLinked to original sources
arXiv2025

A class of Truncated Freud polynomials

Consider the following truncated Freud linear functional $\mathbf{u}_z$ depending on a parameter $z$, $$\langle\mathbf{u}_z,p\rangle=\int_0^\infty p(x)e^{-zx^4}dx,\quad z>0.$$ The aim of this work is to analyze the properties of the sequence of orthogonal polynomials $(P_n)_{n\geq 0}$ with respect to $\mathbf{u}_z$. Such a linear functional is semiclassical and, as a consequence, we get the system of nonlinear difference equations (Laguerre-Freud equations) that the coefficients of the three-term recurrence satisfy. The asymptotic behavior of such coefficients is given. On the other hand, the raising and lowering operators associated with such a linear functional are obtained, and thus a second-order linear differential equation of holonomic type that $(P_n)_{n\geq 0}$ satisfies is deduced. From this fact, an electrostatic interpretation of their zeros is given. Finally, some illustrative numerical tests concerning the behavior of the least and greatest zeros of these polynomials are presented.

math.CA↗
arXiv2017

Matrix biorthogonal polynomials, associated polynomials and functions of the second kind

In this work the interplay between matrix biorthogonal polynomials with respect to a matrix of linear functionals, the $k$-th associated matrix polynomials and the second kind matrix functions, is studied in terms of quasideterminants. A sort of Poincaré's theorem for the ratio of two consecutive matrix functions solutions of a linear difference equation is also presented. Some new formulas connecting these families of matrix functions are given.

math.CA↗
arXiv2016

Christoffel transformations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy

Given a matrix polynomial $W(x)$, matrix bi-orthogonal polynomials with respect to the sesquilinear form $\langle P(x),Q(x)\rangle_W=\int P(x) W(x)\operatorname{d}μ(x)(Q(x))^{\top}$, $P(x),Q(x)\in\mathbb R^{p\times p}[x]$, where $μ(x)$ is a matrix of Borel measures supported in some infinite subset of the real line, are considered. Connection formulas between the sequences of matrix bi-orthogonal polynomials with respect to $\langle \cdot,\cdot\rangle_W$ and matrix polynomials orthogonal with respect to $μ(x)$ are presented. In particular, for the case of nonsingular leading coefficients of the perturbation matrix polynomial $W(x)$ we present a generalization of the Christoffel formula constructed in terms of the Jordan chains of $W(x)$. For perturbations with a singular leading coefficient several examples by Durán et al are revisited. Finally, we extend these results to the non-Abelian 2D Toda lattice hierarchy.

math.CA↗
arXiv2014

Multiple Geronimus transformations

We consider multiple Geronimus transformations and show that they lead to discrete (non-diagonal) Sobolev type inner products. Moreover, it is shown that every discrete Sobolev inner product can be obtained as a multiple Geronimus transformation. A connection with Geronimus spectral transformations for matrix orthogonal polynomials is also considered.

math.CA↗