## seminarios

Este seminario de investigación, puesto en marcha el curso 2002-2003, cuenta con ponencias informales por parte de los miembros del grupo y de investigadores invitados de otras Universidades españolas y extranjeras. La frecuencia habitual es de un seminario al mes, alternativamente en la Universidad de Almerí­a y en la Universidad de Granada, comúnmente con dos sesiones por Seminario, una de las cuales corre preferiblemente a cargo de un profesor invitado.

### CURSO 2021-2022

Jueves, 24 de marzo de 2022
Lugar: Seminario de Matemática Aplicada (2.42) del CITE III, Universidad de Almería.
Hora: 11:00

Joaquín Sánchez Lara,
Título: Compañeros electrostáticos y raíces de polinomios ortogonales
Resumen: 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,
Título: On another characterization of Askey-Wilson polynomials
Resumen: 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.

Ver foto aquí.

Miércoles
, 26 de enero de 2022
Lugar: Seminario de Matemática Aplicada (2.42) del CITE III, Universidad de Almería.
Hora: 12:00

Título: Third degree linear functionals. Applications and open problems.

Resumen: 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.
Título: A local asymptotics for some basic hypergeometric polynomials.

Resumen: The basic $q$-hypergeometric function $_r\phi_s$ is defined by the series
\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}},

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.

### CURSO 2017-2018

Jueves
, 8 de Marzo de 2018
Lugar: Seminario de Matemática Aplicada (2.42) del CITE III, Universidad de Almería.
Hora: 13:00

Roberto Costas Santos, Universidad de Alcalá.
Título: A first study of Zeros of Classical Orthogonal Polynomial.
Resumen: 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.

Ver foto aquí

Jueves
, 30 de Noviembre de 2017
Lugar: Seminario de Matemática Aplicada, Edificio CITE III, 2.42, Universidad de Almería.
Hora: 10:30

Ana B. Castaño Fernández, Universidad de Almería.
Título: Longitudinal vibrations effect via Strehl ratio Optical Transfer Function and Depth of focus.
Resumen: 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.

Título: Polinomios de Appell bivariados. Carácter clásico.
Resumen: 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.

Juan F. Mañas Mañas, Universidad de Almería.
Título: Gegenbauer-Sobolev orthonormal polynomials: Differential operator and eigenvalues.
Resumen: 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].

Referencias

[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-Mañaas, 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.

Ver foto aquí

### CURSO 2016-2017

Jueves
, 9 de Febrero de 2017
Lugar: Seminario de Matemática Aplicada, Edificio CITE III, 2.42, Universidad de Almería.
Hora: 12:00

Título: Orthogonality relations of Al-Salam-Carlitz for general parameters.
Resumen: 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.

Título: Integral functionals of hypergeometric orthogonal polynomials with large parameters and hydrogenic application.
Resumen: 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 RÃ©nyi 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.

Referencias

[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. Enviado.

[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).

Ver foto aquí

### CURSO 2015-2016

Martes
, 14 de Junio de 2016

Hora: 12:30
Dmitry Karp, Far Eastern Federal University, Vladivostok (Rusia).
Título: Distributional G function of Meijer and properties of generalized hypergeometric functions
Resumen: 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.
Hora: 13:30.

Título: Asymptotics of Sobolev orthogonal polynomials on the unit ball
Resumen: 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.

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Viernes, 22 de Abril de 2016

Hora: 11:30
Cleonice F. Bracciali, UNESP - Univ. Estadual Paulista (Brasil).
Título: Orthogonal and para-orthogonal polynomials on the unit circle for measures which are modifications of Lebesgue measure
Resumen: 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.

Hora: 12:30
Juan F. Mañas Mañas, Universidad de Almería.
Título: Asymptotic behavior of varying discrete Sobolev orthogonal polynomials
Resumen: 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.

Ver foto aquí