Papers by Edmundo J. Huertas Cejudo
arXiv (Cornell University), Jan 17, 2013
In this paper we consider the strong asymptotic behavior of Laguerre polynomials in the complex p... more 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 this paper, we propose the use of an alternative expansion, due to Buchholz, in terms of Bessel functions of the first kind. The coefficients in this expansion can be obtained in a straightforward way using symbolic computation. As an application, we derive extra terms in the asymptotic expansion of ratios of Laguerre polynomials in C \ [0, ∞).
Mathematics, 2020
This contribution deals with the sequence {Un(a)(x;q,j)}n≥0 of monic polynomials in x, orthogonal... more 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∈(0,1). We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order q-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by Un(a)(x;q,j), which paves the way to establish an appealing generalization of the so-called J-fractions to the framework of Sobolev-type orthogonality.
Numerical Algorithms
It is well known that Sobolev-type orthogonal polynomials with respect to measures supported on t... more 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 three-term recurrence relation satisfied by the standard sequence of orthonormal polynomials with respect to the 2-iterated Christoffel transformation of the measure.
Journal of Mathematical Analysis and Applications, 2009
Let introduce the Sobolev-type inner product f, g = ∞ 0 f (x)g(x)dµ(x) + M f (0)g(0) + N f (0)g (... more Let introduce the Sobolev-type inner product f, g = ∞ 0 f (x)g(x)dµ(x) + M f (0)g(0) + N f (0)g (0), where dµ(x) = 1 Γ(α + 1) x α e −x dx, M, N ≥ 0, α > −1. In this paper we prove a Cohen type inequality for the Fourier expansion in terms of the orthonormal polynomials associated with the above Sobolev inner product. In particular, for M = N = 0, we extend the result of Markett ([14]) 2000 Mathematics Subject Classification. 42C05, 42C10.
Electronics
In this contribution, we consider the sequence {Hn(x;q)}n≥0 of monic polynomials orthogonal with ... more 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 application for the non-standard cases (when j>0) with respect to the standard classical Krawtchouk case λ=μ=0.
This paper deals with monic orthogonal polynomial sequences (MOPS in short) generated by a Geroni... more This paper deals with monic orthogonal polynomial sequences (MOPS in short) generated by a Geronimus canonical spectral transformation of a positive Borel measure µ, i.e., 1 (x − c) dµ(x) + N δ(x − c), for some free parameter N ∈ R + and shift c. We analyze the behavior of the corresponding MOPS. In particular, we obtain such a behavior when the mass N tends to infinity as well as we characterize the precise values of N such the smallest (respectively, the largest) zero of these MOPS is located outside the support of the original measure µ. When µ is semi-classical, we obtain the ladder operators and the second order linear differential equation satisfied by the Geronimus perturbed MOPS, and we also give an electrostatic interpretation of the zero distribution in terms of a logarithmic potential interaction under the action of an external field. We analyze such an equilibrium problem when the mass point of the perturbation c is located outside the support of µ.
In this paper we consider sequences of polynomials orthogonal with respect to the discrete Sobole... more In this paper we consider sequences of polynomials orthogonal with respect to the discrete Sobolev inner product f, g S = ∞ 0 f (x)g(x)x α e −x dx + F(c)AG(c) t , α > −1, where f and g are polynomials with real coefficients, A ∈ R (2,2) and the vectors F(c), G(c) are A = M 0 0 N , F(c) = (f (c), f ′ (c)) and G(c) = (g(c), g ′ (c)), respectively, with M, N ∈ R + and the mass point c is located inside the oscillatory region for the classical Laguerre polynomials. We focus our attention on the representation of these polynomials in terms of classical Laguerre polynomials and we deduce analyze the behavior of the coefficients of the corresponding five-term recurrence relation when the degree of the polynomials is large enough. Also, the outer relative asymptotics of the Laguerre-Sobolev type with respect to the Laguerre polynomials is analyzed.
En este trabajo se propone un marco didactico en el cual los modelos de ensenanza de la Estadisti... more En este trabajo se propone un marco didactico en el cual los modelos de ensenanza de la Estadistica, Probabilidad y Combinatoria se ven reflejados a partir de un conjunto de actividades ludicas que acercan al alumnado al estudio matematico de los fenomenos aleatorios relacionados con el azar desde un entorno mas experimental y divertido. La propuesta de este trabajo se fundamenta teoricamente en los conceptos basicos de combinatoria, estadistica y probabilidad utilizando en ocasiones el ordenador como herramienta de trabajo para facilitar la asignacion de probabilidades a sucesos aleatorios, especialmente cuando hay que recurrir a tecnicas combinatorias para efectuar recuentos. En definitiva, intentamos fomentar el interes por los fenomenos aleatorios en general, presentes en nuestra vida cotidiana, a traves del juego, lo que anima a los estudiantes a poner mas empeno en el estudio de materias a priori no interesantes o con excesiva dificultad para ellos.
Excess of deaths is a technique used in epidemiology to assess the deaths caused by an unexpected... more 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 apply the results obtained to evaluate the evolution of the present pandemic in Brazil and Spain, which allows in particular to compare how well (or bad) these countries have managed the pandemic. For Brazil, our calculations refute the claim made by some officials that the present pandemic is “a little flu”. Some studies suggest that the virus could be lying dormant across the world before been detected for the first time. In that regard, our results show that there is no evidence of deaths by the virus in 2019.
The q-Hermite I-Sobolev type polynomials of higher order are consider for their study. Their hype... more 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-Sobolev type polynomials of higher order are also established.
In this contribution, the behavior of zeros of orthogonal polynomials associated with canonical l... more In this contribution, the behavior of zeros of orthogonal polynomials associated with canonical linear spectral transformations of measures supported in the real line is analyzed. An electrostatic interpretation of them is given.
In this contribution we complete and deepen the results in (Huertas et al, Proc Am Math Soc 142(5... more In this contribution we complete and deepen the results in (Huertas et al, Proc Am Math Soc 142(5), 1733–1747, 2014, [1]) by introducing a determinantal form for the ladder operators concerning the infinite sequence \(\{Q_{n}(x)\}_{n\ge 0}\) of monic polynomials orthogonal with respect to the following Laguerre–Krall inner product $$\begin{aligned} \langle f,g\rangle _{\nu } =\int _{0}^{+\infty }f(x)g(x)x^{\alpha } e^{-x}dx+\sum _{j=1}^{m}a_{j}\,f(c_{j})g(c_{j}), \end{aligned}$$ where \(c_{j}\in \mathbb {R}_{-} \cup \{0\}\). We obtain for the first time explicit formulas for these ladder (creation and annihilation) operators, and we use them to obtain several algebraic properties satisfied by \(Q_{n}(x)\). As an application example, based on the structure of the above inner product \(\langle f,g\rangle _{\nu } \), we consider a convex linear combination of continuous and discrete measures that leads to establish an interesting research line concerning the Laguerre–Krall polynomials ...
Symmetry
In this contribution we obtain some algebraic properties associated with the sequence of polynomi... more 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 relation, a three-term recurrence relation with rational coefficients, and an explicit expression for its norms. Moreover, we use these results to deduce asymptotic properties for the recurrence coefficients and a nonlinear difference equation that they satisfy, in the particular case when dμ(x)=e−x4dx.
Mathematics
In this contribution, we propose an algorithm to compute holonomic second-order differential equa... more 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-order differential equations whose solutions are orthogonal polynomials associated with some spectral transformations of a measure on the unit circle, as well as orthogonal polynomials associated with coherent pairs of measures on the unit circle.
We analyze the effect of symmetrization in the theory of multiple orthogonal polynomials. For a s... more 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 arbitrary sequence of type II multiple orthogonal polynomials with respect to a set of d linear functionals, we obtain a total of d+1 sequences of type II multiple orthogonal polynomials, which can be used to construct a new sequence of symmetric type II multiple orthogonal polynomials. Finally, we prove a Favard-type result for certain sequences of matrix multiple orthogonal polynomials satisfying a matrix four-term recurrence relation with matrix coefficients.
Sets of orthogonal martingales are importants because they can be used as stochastic integrators ... more 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 orthogonalization procedure of a sequence of martingales related to a certain Lévy process, called Teugels Martingales. Since the Uvarov transformation depends by a c<0, we are able to provide infinite sets of strongly orthogonal martingales, each one for every c in (-infty,0). In a similar fashion of [21], we introduce a suitable isometry between the space of polynomials and the space of linear combinations of Teugels martingales as well as the general orthogonalization procedure. Finally, the new construction is applied to the Gamma process.
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Papers by Edmundo J. Huertas Cejudo