Edmundo J. Huertas Cejudo

University of Alcalá | UAH · Department of Physics and Mathematics

This paper deals with monic orthogonal polynomials generated by a Geronimus canonical spectral transformation of the Laguerre classical measure: \[\frac{1}{x-c}x^{\alpha }e^{-x}dx+N\delta (x-c), \] for $x\in[0,\infty)$, $\alpha>-1$, a free parameter $N\in \mathbb{R}_{+}$ and a shift $c<0$. We analyze the asymptotic behavior (both strong and relativ...

We analyze the effect of symmetrization in the theory of multiple orthogonal polynomials. For a symmetric sequence of type II multiple orthogonal polynomials satisfying a high-term recurrence relation, we fully characterize the Weyl function associated to the corresponding block Jacobi matrix as well as the Stieltjes matrix function. Next, from an...

In this paper we consider sequences of polynomials orthogonal with respect to certain discrete Laguerre-Sobolev inner product, with two perturbations (involving derivatives) located inside the oscillatory region for the classical Laguerre polynomials. We focus our attention on the representation of these polynomials in terms of the classical Laguer...

In this paper we consider the strong asymptotic behavior of Laguerre polynomials in the complex plane. The leading behavior is well known from Perron and Mehler-Heine formulas, but higher order coefficients, which are important in the context of Krall-Laguerre or Laguerre-Sobolev-type orthogonal polynomials, are notoriously difficult to compute. In...

The aim of the work is to construct new polynomial systems, which are solutions to certain functional equations which generalize the second-order differential equations satisfied by the so called classical orthogonal polynomial families of Jacobi, Laguerre, Hermite and Bessel. These functional equations can be chosen to be of different type: fracti...

It is well known that Sobolev-type orthogonal polynomials with respect to measures supported on the real line satisfy higher-order recurrence relations and these can be expressed as a (2N + 1)-banded symmetric semi-infinite matrix. In this paper, we state the connection between these (2N + 1)-banded matrices and the Jacobi matrices associated with...

It is well known that Sobolev-type orthogonal polynomials with respect to measures supported on the real line satisfy higher-order recurrence relations and these can be expressed as a (2N+1)-banded symmetric semi-infinite matrix. In this paper we state the connection between these (2N+1)-banded matrices and the Jacobi matrices associated with the t...

In this contribution, we consider the sequence {Hn(x;q)}n≥0 of monic polynomials orthogonal with respect to a Sobolev-type inner product involving forward difference operators For the first time in the literature, we apply the non-standard properties of {Hn(x;q)}n≥0 in a watermarking problem. Several differences are found in this watermarking appli...

Excess of deaths is a technique used in epidemiology to assess the deaths caused by an unexpected event. For the present COVID–19 pandemic, we discuss the performance of some linear and nonlinear time series forecasting techniques widely used for modeling the actual pandemic and provide estimates for this metric from January 2020 to April 2021. We...

The q-Hermite I-Sobolev type polynomials of higher order are consider for their study. Their hypergeometric representation is provided together with further useful properties such as several structure relations which give rise to a three-term recurrence relation of their elements. Two different q-difference equations satisfied by the q-Hermite I-So...

In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:p,qs=∫Rp(x)q(x)dμ(x)+M0p(0)q(0)+M1p′(0)q′(0), where p,q are polynomials, M0, M1 are non-negative real numbers and μ is a symmetric positive measure. These include a five-term recurrence relat...

We analyze the effect of symmetrization in the theory of multiple orthogonal polynomials. For a symmetric sequence of type II multiple orthogonal polynomials satisfying a high-term recurrence relation, we fully characterize the Weyl function associated to the corresponding block Jacobi matrix as well as the Stieltjes matrix funcion. Next, from an a...

The present volume contains the Proceedings of the Seventh Iberoamerican Workshop in Orthogonal Polynomials and Applications (EIBPOA, which stands for Encuentros Iberoamericanos de Polinomios Ortogonales y Aplicaciones, in Spanish), held at the Universidad Carlos III de Madrid, Leganés, Spain, from July 3 to July 6, 2018. These meetings were mainly...

This contribution deals with the sequence {Un(a)(x;q,j)}n≥0 of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number j of q-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number q∈...

This contribution deals with the sequence $\{\mathbb{U}_{n}^{(a)}(x;q,j)\}_{n\geq 0}$ of monic polynomials, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam--Carlitz I orthogonal polynomials, and involving an arbitrary number of $q$-derivatives on the two boundaries of the corresponding orthogonality interval. We prov...

In this contribution, we propose an algorithm to compute holonomic second-order differential equations satisfied by some families of orthogonal polynomials. Such algorithm is based in three properties that orthogonal polynomials satisfy: a recurrence relation, a structure formula, and a connection formula. This approach is used to obtain second-ord...

In this contribution, we consider the sequence {Qnλ}n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{Q_{n}^{\lambda }\}_{n\geq 0}$\end{document} of monic polynomials...

In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product \begin{equation*} \left\langle p,q\right\rangle _{s}=\int_{\mathbb{R}}p(x)q(x)e^{-x^{4}}dx+M_{0}p(0)q(0)+M_{1}p^{\prime }(0)q^{\prime }(0), \end{equation*}% where $p,q$ are polynomials, $M_{0}$ and $M_{1}$ are nonnegative...

In this contribution we consider sequences of monic polynomials orthogonal with respect to the standard Freud-like inner product involving a quartic potential ⟨p,q⟩M=∫Rp(x)q(x)e−x4+2tx2dx+Mp(0)q(0). We analyze some properties of these polynomials, such as the ladder operators and the holonomic equation that they satisfy and, as an application, we...

Let consider the Sobolev type inner product <f, g> =∫ f(x)g(x)dμ(x) + Mf(c)g(c) + Nf'(c)g'(c), where dμ(x) = (x^α)exp{-x}dx, a > -1, is the Laguerre measure, c < 0, and M, N >= 0. In this paper we get a Cohen-type inequality for Fourier expansions in terms of the orthonormal polynomials associated with the above Sobolev inner product. Then, as an i...

This paper deals with monic orthogonal polynomial sequences (MOPS in short) generated by a Geronimus canonical spectral transformation of a positive Borel measure $\mu$, i.e., \begin{equation*} \frac{1}{(x-c)}d\mu (x)+N\delta (x-c), \end{equation*} for some free parameter $N \in \mathbb{R}_{+}$ and shift $c$. We analyze the behavior of the corres...

In this paper we survey how an inner product derived from an Uvarov transformation of the Laguerre weight function is used in the orthogonalization procedure of a sequence of martingales related to a Lévy process. The orthogonalization is done by isometry and it is based in previous works of Nualart and Schoutens (see [18] and [19]), where the resu...

Resumen: En el presente trabajo se realiza un análisis de la presencia de la acción tutorial en el ámbito universitario. Partiendo de un repaso de las diferentes modalidades de tutorización, y de las competencias y funciones que el profesor-tutor desempeña en el marco de los estudios superiores, se le proponen al profesor-tutor una serie de herrami...

In this paper we consider the sequences of polynomials orthogonal with respect to the Laguerre measure modified by m Dirac mass points located in the negative real semiaxis. We first focus our attention in the representation of these polynomials in terms of the standard Laguerre polynomials. Next we find the explicit formula for their outer relativ...

Sets of orthogonal martingales are importants because they can be used as stochastic integrators in a kind of chaotic representation property, see [20]. In this paper, we revisited the problem studied by W. Schoutens in [21], investigating how an inner product derived from an Uvarov transformation of the Laguerre weight function is used in the orth...

In this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product $$ \left\langle p,q\right\rangle _{S}=\int_{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{\prime }(a)q^{\prime }(a),\alpha >-1 $$where N ∈ ℝ + , and a ∈ ℝ − . We study the outer relative asymptotics of the...

This thesis is focused on the so called standard and nonstandard families of orthogonal polynomials. We begin by giving a brief historical introduction and some preliminary concepts about orthogonal polynomial sequences. The first part of the memoir is devoted to analyze the zeros and some outer asymptotic properties of the so called Krall-type ort...

It is known that some orthogonal systems are mapped onto other orthogonal systems by the Fourier transform. In this article we introduce a finite class of orthogonal functions, which is the Fourier transform of Routh–Romanovski orthogonal polynomials, and obtain its orthogonality relation using Parseval identity.

In this paper we deal with sequences of polynomials orthogonal with respect to the discrete Sobolev inner product <f,g>S = 0 ω(x)f(x)g(x)dx + M f(ξ)g(ξ)+ N f(ξ)g(ξ), " class="math-display" /> where ω is a weight function, ξ ≤ 0, and M,N ≥ 0. The location of the zeros of discrete Sobolev orthogonal polynomials is given in terms of the zeros of stand...

In this paper we analyze the behaviour of the zeros of polynomials orthogonal with respect to the Uvarov perturbation of a positive Borel measure $d\mu$. When the measure is semiclassical, then its electrostatic interpretation is given.

In this paper, we consider sequences of monic polynomials orthogonal with respect to an inner product [image omitted] where M+, and a-. We focus our attention on the representation of these polynomials in terms of the standard Laguerre polynomials as well as hypergeometric functions. The lowering and raising operators associated with these polynomi...

In this contribution, we study the analytic properties of orthogonal polynomials associated with the Uvarov's canonical linear spectral transformations on the Laguerre's classical measure supported in the real line. The distribution of their zeros is analyzed in terms of their dependence of N , the weight of the discrete part of the perturbed measu...