seminars


This research seminar, that started during the 2002-2003 academic year, hosts informal talks by both members of the team and invited researchers from other Spanish and foreign universities. The customary frequency is once a month, alternatively at the Unversity of Almería and Granada, and each seminar usually has two one-hour sessions, one of them preferably by a visitor.


Year 2022-2023


Friday, February 3, 2023
Room: Seminario Paul Erdös (seminar 2.42) from building CITE III, Universidad de Almería.
Time: 11:00.

Alberto Lastra, Universidad de Alcalá (Spain).
Title: Una primera aproximación a las soluciones de sistemas de ecuaciones momento diferenciales.
Abstract: En el seminario se motivará e introducirá el estudio de sistemas de ecuaciones momento-diferenciales, generalizando sistemas de EDOs, de ecuaciones diferenciales fraccionarias y ecuaciones en q-diferencias, dependiendo de la elección de la sucesión de momentos asociada a una elección concreta de medidas de tipo Laguerre. Se determinará la solución general de estas a partir de núcleos para la sumabilidad y se establecerán ciertas propiedades.

Edmundo J. Huertas, Universidad de Alcalá (Spain).
Title: Sobre una perspectiva matricial de las perturbaciones tipo Sobolev-discreto de medidas.
Abstract: Es bien sabido que los polinomios ortogonales de tipo Sobolev con respecto a medidas soportadas en la recta real satisfacen relaciones de recurrencia de orden superior, y que éstas pueden expresarse como una matriz semi-infinita simétrica de (2N + 1) bandas. En este trabajo se establece la conexión entre estas matrices de (2N+1)-banda y las matrices de Jacobi asociadas con la relación de recurrencia de tres términos satisfecha por la familia de polinomios ortonormales con respecto a la doble perturbación canónica de Christoffel de la medida considerada.
Este seminario está basado en la reciente publicación: Carlos Hermoso, Edmundo J. Huertas, Alberto Lastra, and Francisco Marcellán, Higher-order recurrence relations, Sobolev-type inner products and matrix factorizations, Numer. Algorithms, 92 (2023), 665-692, DOI.

See photos: Photo 1 Photo 2.

Friday, December 16, 2022
Room: Seminario Paul Erdös del Edificio CITE III, Universidad de Almería.
Time: 11:00.

Maria das Neves Rebocho, Universidade da Beira Interior (Portugal).
Title: Deformed orthogonal polynomials on the real line.
Abstract: The construction of families of orthogonal polynomials on the real line from a given system of orthogonal polynomials or from a given modified orthogonalizing measure has been subject of many studies. Well-known connections with integrable systems, Painlevé equations (discrete and continuous), random matrices, and many other topics from the literature of Mathematical-Physics have been recently studied.
In this talk we focus on the so-called Laguerre-Hahn class on the real line, that is, the sequences of orthogonal polynomials whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients. We shall take Stieltjes functions subject to a deformation parameter, $t$, and we derive systems of differential equations and give Lax pairs, yielding non-linear differential equations in $t$ for the recurrence relation coefficients and Lax matrices of the orthogonal polynomials. A specialisation to a non semi-classical case obtained via a M\"{o}bius transformation of a Stieltjes function related to a modified Jacobi weight is studied in detail, showing that such a system is governed by a differential equation of the Painlevé type P$_\textrm{VI}$.

Teresa E. Pérez, Universidad de Granada (Spain).
Title: Operadores de tipo Bernstein y polinomios de Jacobi .
Abstract: Los polinomios de Bernstein fueron introducidos por S. Bernstein en 1912 para proporcionar una demostración constructiva del teorema de aproximación de Weierstrass, que afirma que toda función continua definida en el intervalo $[0, 1]$ puede ser aproximada uniformemente por polinomios de Bernstein.
A lo largo del tiempo se han modificado los operadores de Bernstein para que cumplan propiedades adicionales, como la preservación de la derivación o el hecho de poseer familias completas de funciones propias. En este caso estudiaremos operadores de tipo Bernstein que poseen los polinomios ortogonales de Jacobi como funciones propias y, además, preservan derivadas en el sentido de que la derivada del operador aplicada a una función coincide con un operador de familia adyacente aplicado a la derivada de la función.
Este es un trabajo conjunto con David Lara.

See photo here (Darío Ramos López also attended the seminar, but the picture was taken later).

YEAR 2021-2022



 Friday, July 22, 2022
Room: Seminario Paul Erdös del Edificio CITE III, Universidad de Almería.

Veronika Pillwein, Johannes Kepler University Linz (Austria).
Time: 10:00.
Title: Symbolic methods for special functions.
Abstract: The beginning of computer algebra is dated by Wikipedia to around 1970: Bruno Buchberger's seminal PhD thesis, in which he developed Groebner bases, was published in 1965, in 1975 George E. Collins introduced the notion of Cylindrical Algebraic Decomposition for doing quantifier elimination over the real numbers, and in 1982 the Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm was invented. These three methods are often referred to as the pillars of symbolic computation. In the late 1980s, early 1990s, Doron Zeilberger introduced the holonomic systems approach for special functions identities. Since then, these algorithms have been extended and implemented and nowadays even the non-expert in computer algebra has a variety of methods to choose from. In my research, several of these have been very handy in proving and discovering identities and inequalities (in particular) on orthogonal polynomials. In this talk, I want to give an overview on different applications and share how one can include symbolic methods in daily calculations.

Diego E. Dominici, Johannes Kepler University Linz (Austria).
Time: 11:00.
Títle: Comparative asymptotics for discrete semiclassical orthogonal polynomials.
Abstract: We study the ratio ((P_{n}(x;z))/(φ_{n}(x))) asymptotically as n→∞, where the polynomials P_{n}(x;z) are orthogonal with respect to a discrete linear functional and φ_{n}(x) denote the falling factorial polynomials.
We give recurrences that allow the computation of high order asymptotic expansions of P_{n}(x;z) and give examples for most discrete semiclassical polynomials of class s.
We show several plots illustrating the accuracy of our results.
  

See photo here (María Inmaculada López García, Darío Ramos López, Dieudonne Mbouna, Miguel Ángel Sánchez Granero and Joaquín Sánchez Lara also attended the seminar, but the picture was taken later).


Thursday, March 24,  2022
Room: Seminario de Matemática Aplicada (2.42), CITE III, University of Almería.
Time: 11:00

Joaquín Sánchez Lara, University of Granada.
Title: Compañeros electrostáticos y raíces de polinomios ortogonales
Abstract: La interpretación electrostática de las raíces de polinomios ortogonales es una de las descripciones más elegantes y útiles que existen a la hora de estudiar la distribución que éstas siguen así como su asintótica para grados altos. Sin embargo esta interpretación es bien conocida solamente en el caso de polinomios ortogonales clásicos y solo de manera relativa en otros casos. El objetivo de esta charla es mostrar una construcción que permite obtener dicha interpretación electrostática partiendo solamente del polinomio (no necesariamente ortogonal) y un peso semiclásico. En esta construcción aparece un segundo polinomio, el compañero electrostático,  cuyas raíces son fundamentales en el modelo electrostático que se propone. Si bien esta construcción es general, prestaremos especial atención al caso de polinomios ortogonales múltiples de tipo II, en donde el modelo resultante permite explicar de forma unificada las diferentes descripciones asintóticas que aparecen en la literatura.  

Dieudonne Mbouna, University of Almería.
Title: On another characterization of Askey-Wilson polynomials
Abstract: In this talk we expose the theory of classical orthogonal polynomials on lattices and we use this to give a Al-Salam and Chihara type characterization of classical orthogonal polynomials on lattices.

 

See photo here.


Wednesday, January 26, 2022
Room: Seminario de Matemática Aplicada (2.42) del CITE III, Universidad de Almería.
Time: 12:00

Francisco Marcellán Español, Universidad Carlos III de Madrid.
Title: Third degree linear functionals. Applications and open problems.
Abstract: In this talk we will focus the attention on linear functionals such that the corresponding Stieltjes function satisfies a cubic equation with polynomial coefficients. They are called third degree linear functionals. Some algebraic properties of these functionals are discussed. They belong to the Laguerre-Hahn class and several examples are presented. Finally, some open problems will be stated.

Juan Francisco Mañas Mañas, Universidad de Almería.
Title: A local asymptotics for some basic hypergeometric polynomials.
Abstract: The basic $q$-hypergeometric function $ _r\phi_s$ is defined by the series
\begin{equation}\label{q-basic}
_r\phi_{s}\left(\begin{array}{l}
a_{1}, \ldots, a_{r} \\
b_{1}, \ldots, b_{s}
\end{array} ; q, z\right)
=\sum_{k=0}^{\infty} \frac{\left(a_{1}; q\right)_{k}\cdots\left(a_{r} ; q\right)_{k}}{\left(b_{1}; q\right)_{k}\cdots\left(b_{s} ; q\right)_{k}}\left((-1)^kq^{\binom{k}{2}}\right)^{1+s-r}\frac{z^{k}}{(q ; q)_{k}},
\end{equation}
where $0<q<1$ and $ \left(a_{j}; q\right)_{k}$ and $\left(b_{j}; q\right)_{k}$ denote the $q$-analogues of the Pochhammer symbol.

When one of the parameters $a_j $ in (\ref{q-basic}) is equal to $q^{-n}$ the basic $q$-hypergeometric function is a polynomial of degree at most $n$ in the variable $z$. Our objective is to obtain a type of local asymptotics, known as Mehler--Heine asymptotics, for $q$-hypergeometric polynomials when $r=s$.

Concretely, by scaling adequately these polynomials we intend to get a limit relation between them and a $q$-analogue of the Bessel function of the first kind. Originally, this type of local asymptotics was introduced for Legendre orthogonal polynomials (OP) by the German mathematicians H. E. Heine and G. F. Mehler in the 19th century. Later, it was extended to the families of classical OP (Jacobi, Laguerre, Hermite), and more recently, these formulae were obtained for other families as discrete OP, generalized Freud OP, multiple OP or Sobolev OP, among others.

These formulae have a nice consequence about the scaled zeros of the polynomials, i.e. using the well--known Hurwitz's theorem we can establish a limit relation between these scaled zeros and the ones of a Bessel function of the first kind. In this way, we are looking for a similar result in the context of the $q$-analysis and we will illustrate the results with numerical examples.

This is a joint work with Juan J. Moreno-Balcázar.

See photo here.



YEAR 2017-2018



 Thursday, March 8, 2018
Venue: Seminario de Matemática Aplicada (2.42) del CITE III, Universidad de Almería.
Time: 13:00

Roberto Costas Santos, Universidad de Alcalá.
Title: A first study of Zeros of Classical Orthogonal Polynomial.
Abstract: In this short work we consider some basic results connected with the zeros of classical orthogonal polynomials (COP). Given a orthogonal polynomial sequence (OPS)m namely $(p_n)_{n\ge 0}$, we set a positive integer $N$ and we obtain for such value a new set of orthogonality properties for such for $(p_n)_{n\ge 0}$, we obtain closed expressions for $p_n(x)$ for $n\le N$ and the Hankel matrix for such polynomials in terms of the zeros of $p_N(x)$.

With these results we believe we are able to establish some conditions on the parameters of the family. We also prove that the derivative of every COP is the kernel of itself. Explicit expressions are given.

See photo here


Thursday, November 30, 2017
Venue: Seminario de Matemática Aplicada, Edificio CITE III, 2.42, Universidad de Almería.
Time: 10:30

Ana B. Castaño Fernández, Universidad de Almería.
Title: Longitudinal vibrations effect via Strehl ratio Optical Transfer Function and Depth of focus.
Abstract: This work consist on computing and analysing the Strehl Ratio Optical Transfer Function Metric (SOTF) from the wavefront. For that, we search some efficient algorithms for calculating the SOTF metric, especially for radially-symmetric wavefronts. It is an alternative for the standard approach via the Discrete Fourier Transform (DFT), developing integral formulas that lack the oscillatory character and can be evaluated accurately by quadratures. Therefore, numerical analysis of scenarios, studied previously empirically, when longitudinal vibrations are present, confirming an improvement of the visual quality expressed by the averaged SOTF metric, where the only two possible averaging were considered: in the diffraction integral and in the Point Spread Function (PSF). For finishing, we study changes in the objective depth of focus by means of a Through-focus Analysis using the averaged SOTF metric.

A similar study was carried out for the modified metric, called the visual Strehl ratio calculated from the optical transfer function (VSOTF) which takes into account also the Neural Contrast Sensitivity Function (NCSF) of the eye. It is regarded as a more accurate metric of the visual quality. We perform also a comparison of our conclusions for both the SOTF and the VSOTF.

Fátima Lizarte López, Universidad de Granada.
Title: Polinomios de Appell bivariados. Carácter clásico.
Abstract: Los polinomios de Appell, introducidos por el propio Paul Appell en 1881, se definen a partir de una fórmula de tipo Rodrigues (utilizando operadores de derivación sucesiva en varias variables sobre una función determinada) como una extensión no trivial a dos variables de los polinomios clásicos de Jacobi que son ortogonales en el intervalo [0, 1].

En este trabajo se estudia en profundidad dichos polinomios y sus propiedades más importantes, haciendo un recorrido desde las propiedades más conocidas hasta las que se han obtenido recientemente.

Demostramos la ortogonalidad de estos polinomios con respecto a un producto escalar definido sobre el triángulo unidad mediante una función peso dependiente de tres parámetros, que es la extensión de la función peso de los polinomios de Jacobi al caso bivariado. También estudiamos su expresión hipergeométrica.

A continuación, hacemos un estudio en profundidad del carácter clásico de los polinomios de Appell; se suelen llamar clásicos porque son funciones propias de un operador diferencial en derivadas parciales de segundo orden cuyos coeficientes son polinomios de grado fijo, y están relacionados con la función peso considerada, y que le confiere su carácter hipergeométrico. Pero, además satisfacen otras propiedades diferenciales menos conocidas: ecuación de tipo Pearson para la función peso, fórmula de estructura, ortogonalidad de los gradientes, ortogonalidad de tipo Sobolev, entre otras.

Juan F. Mañas Mañas, Universidad de Almería.
Title: Gegenbauer-Sobolev orthonormal polynomials: Differential operator and eigenvalues.
Abstract: In this talk, we consider the following discrete Sobolev inner product involving the Gegenbauer weight $$(f,g)_S:=\int_{-1}^1f(x)g(x)(1- x^2)^{\alpha}dx+M\big[f^{(j)}(-1)g^{(j)}(-1)+f^{(j)}(1)g^{(j)}(1)\big],$$ where $\alpha>-1,$ $j\in \mathbb{N}\cup \{0\},$ and $M>0.$ Let $\{q_n^{(\alpha,M,j)}\}_{n\geq0}$ be the sequence of orthonormal polynomials with respect to the above inner product. These polynomials are eigenfunctions of a differential operator $\mathbf{T} $ (see [1]). We establish the asymptotic behavior of the corresponding eigenvalues which we denote by $\widetilde{\lambda}_n$.

Finally, following the techniques of [3] and [4], we deduce the Mehler-- Heine type asymptotics for $\{q_n^{(\alpha,M,j)}\}_{n\geq0}$ and the location of the zeros of this polynomials.

Joint work with Lance L. Littlejohn (Baylor University, United States), Juan J. Moreno-Balcázar (Universidad de Almería, Spain), and Richard Wellman (Westminster College SLC, United States), see [2].

References

[1] H. Bavinck, J. Koekoek, Differential operators having symmetric orthogonal polynomials as eigenfunctions, J. Comput. Appl. Math. 106 (1999), 369-393.

[2] L. L. Littlejohn, J. F. Mañas-Mañas, J. J. Moreno-Balcázar, R. Wellman, Differential operator for discrete Gegenbauer-Sobolev orthogonal polynomials: eigenvalues and asymptotics, submitted (arXiv:1705.08167).

[3] J. F. Mañas-Mañas, F. Marcellán, J. J. Moreno-Balcázar, Asymptotics for varying discrete Sobolev orthogonal polynomials, Appl. Math. Comput. 314 (2017), 65-79.

[4] A. Peña, M. L. Rezola, Connection formulas for general discrete Sobolev polynomials: Mehler-Heine asymptotics, Appl. Math. Comput. 261 (2015), 216-230.

See photo here



year 2016-2017



Thursday, February 9, 2017
Venue: Seminario de Matemática Aplicada, Edificio CITE III,2.42, Universidad de Almería.
Time: 12:00

Roberto Costas, Universidad de Alcalá.
Title: Orthogonality relations of Al-Salam-Carlitz for general parameters.
Abstract: In this talk we describe the orthogonality conditions satisfied by Al-Salam-Carlitz polynomials $U^{(a)}_n(x;q)$ when the parameters a and q are not necessarily real nor 'classical', i.e., the linear functional u with respect to such polynomial sequence is quasi-definite and not positive definite. We establish orthogonality on a simple contour in the complex plane which depends on the parameters. In all cases we show that the orthogonality conditions characterize the Al-Salam-Carlitz polynomials $U_n^{(a)}(x;q)$ of degree $n$ up to a constant factor. We also obtain a generalization of the unique generating function for these polynomials.

This is a joint work with Howard S. Cohl, and Wenqing Xu.

Irene V. Toranzo, Universidad de Granada.
Title: Integral functionals of hypergeometric orthogonal polynomials with large parameters and hydrogenic application.
Abstract: The analytical determination of the $\alpha \rightarrow \infty $ asymptotic behavior of various power and logarithmic integral functionals of Laguerre ($L_{m}^{\alpha} (x)$) and Gegenbauer ($C_{m}^{(\alpha}) (x)$) polynomials has allowed us to calculate the uncertainty measures of Heisenberg and entropic types for $D$-dimensional hydrogenic and harmonic quantum systems. They are given by the radial expectation values and the Rényi entropies of the probability density of the quantum-mechanical states of these systems. In this work we have used the known asymptotics of the Laguerre and Gegenbauer polynomials together with some Hermite-type expansions of the entropy-like functionals of these polynomials [1], as well as two novel approaches for the computation of the leading term of the hypergeometric functions involved in the large $D$ limit of the radial expectation values [2]. Then, these quantities are used to obtain generalizations of the standard Heisenberg-like and logarithmic uncertainty relations, and some upper and lower bounds to the entropic uncertainty measures (Shannon, Rényi, Tsallis) of the $D$-dimensional hydrogenic system.

References

[1] I. V. Toranzo, N. M. Temme and J. S. Dehesa, Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters. J. Phys. A: Math. Theor. Submitted.

[2] I. V. Toranzo, A. Martínez-Finkelshtein and J. S. Dehesa, Heisenberg- like uncertainty measures for D-dimensional hydrogenic systems at large D. J. Math. Phys. 57, 082109 (2016).

See photo here

year 2015-2016



Tuesday, June 14, 2016
Venue: Seminario de Matemática Aplicada, Edificio CITE III, Universidad de Almería.

Time: 12:30
Dmitry Karp, Far Eastern Federal University, Vladivostok (Rusia).
Title: Distributional G function of Meijer and properties of generalized hypergeometric functions
Abstract: We discuss various new properties of a particular case of Meijer's G-function, including integral and functional equations, nonnegativity conditions and number of zeros, convergence of measures with G-function density and regularization of integrals containing G-function. Some of these properties are then applied to derive new representations for generalized hypergeometric functions and establish some new and old facts about them. In particular, for the generalized hypergeometric functions of Bessel type we find some positivity conditions, inequalities and information about zeros of this function. 
Time: 13:30
Antonia Mª Delgado, Universidad de Granada.
Title: Asymptotics of Sobolev orthogonal polynomials on the unit ball
Abstract: Sobolev orthogonal polynomials on the unit ball are studied The corresponding Sobolev inner product is defined involving outward normal derivatives on the sphere. We will give explicit representation for orthogonal polynomials and reproducing kernels in term of classical polynomials on the ball. From these explicit expressions, algebraic properties and asymptotic behaviour of Christoffel functions will be deduced.

See photo here

Friday, April 22, 2016
Venue: Seminario de Matemática Aplicada, Edificio CITE III, Universidad de Almería.

Time: 11:30
Cleonice F. Bracciali, UNESP - Univ. Estadual Paulista (Brasil).
Title: Orthogonal and para-orthogonal polynomials on the unit circle for measures which are modifications of Lebesgue measure
Abstract: We consider nontrivial probability measures, obtained as simple modifications of the Lebesgue measure, which include mass points at $z=1$ and $z=i$. The orthogonal polynomials on the unit circle (OPUC), the para-orthogonal polynomials and Toeplitz matrices associated with these measures are presented, through explicit formulas for the Verblunsky coefficients.

Joint work with Jairo S. Silva and A. Sri Ranga.

Time: 12:30
Juan F. Mañas Mañas, Universidad de Almería.
Title: Asymptotic behavior of varying discrete Sobolev orthogonal polynomials
Abstract: We consider a varying discrete Sobolev inner product involving a finite positive Borel measure $\mu$ supported on a infinite subset of the real line. This talk is devoted to study the asymptotic properties of the orthonormal polynomials with respect to the Sobolev inner product

$$(f,g)_S=\int f(x)g(x)d \mu+M_nf^{(j)}(c)g^{(j)}(c),$$

where $c\in \mathbb{R}$, $j \geq 0$ and $\{M_n\}_n$ is a sequence of nonnegative real numbers. We are interested in Mehler--Heine type formulae because they describe the asymptotic behavior of these polynomials around point $c$, where we have located the perturbation. Moreover, asymptotic properties of the zeros of these Sobolev polynomials in terms of the zeros of other special functions are provided.

Joint work with F. Marcellán and J.J. Moreno-Balcázar.

See photo here



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