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Weekly Seminar
 WS Season 2017/2018

# Title:  Random polynomials satisfying a three-term recurrence relation

Abstract: In this talk we consider polynomials Pn(z) satisfying a three-term recurrence relation of the form zPn=Pn+1+an Pn-1, with positive random coefficients an. Assuming the coefficients an are i.i.d., we study the mean zero asymptotic distribution and mean Padé asymptotic distribution of these polynomials, as well as relations between them. This is a joint work with V. Prokhorov.

# Title: Discrete Laguerre Sobolev orthogonal polynomials.

Abstract: In this talk we consider the sequence of discrete Laguerre Sobolev orthogonal polynomials {Sn}n=1∞ with an arbitrary (finite) number of mass points. This is the monic sequence of orthogonal polynomials with respect to an inner product in the form

<f,g>=∫0f(x)g(x) ωα(x)dx+∑i=0mj=1Ni λi,j f(i)(ci,j) g(i)(ci,j).

where m, Ni Î Z+ , ci,j <0 for  i=0, … ,m and j=1, … , Ni. We obtain interlacing properties of zeros of this sequence and we prove that Sn has at least n-d zeros on (0,∞), where is a constant depending only of the mass points ci,j, which is equal to d=i=0m(i+1)Ni in the case of the mass points ci,j are all different. In the above case an explicit formula for the outer relative asymptotic of these polynomials is also described.

# Speaker: Irene Valero Toranzo. Departamento de Física Atómica, Molecular y Nuclear de la Universidad de Granada.

Title:  Parametric asymptotics of Laguerre and Gegenbauer integral functionals and pseudoclassical applications

Abstract: In this work we study the parametric asymptotic behavior (α → ∞) of some power and logarithmic integral functionals [1] of the Laguerre L(α)m(x) and Gegenbauer C(α)m(x) polynomials. These orthogonal polynomials control the wavefunctions of the D-dimensional hydrogenic and oscillator-like systems in both position and momentum spaces, where the parameter α depends linearly on D. To do so, we design a mathematical procedure which makes use of the generalized Laguerre and Gegenbauer’s series representation, jointly with various general results of asymptotics. Then, we show novel Hermite-type expansions for the Laguerre and Gegenbauer polynomials which have mathematical interest per se. Finally, these results are applied to calculate the main entropic uncertainty measures (Fisher information, Shannon and Rényi entropies) for the high-dimensional or pseudoclassical states of the quantum-mechanical systems mentioned above [2].

References

[1] N. M. Temme, I. V. Toranzo and J. S. Dehesa, Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters. J. Phys. A: Math. Theor. 50(21), 215206 (2017).

[2] D. Puertas-Centeno, N. M. Temme, I. V. Toranzo, and J. S. Dehesa, Entropic uncertainty measures for large dimensional hydrogenic systemsJ. Math. Phys. 58, 103302 (2017).

Thursday, February 22,  2018, 17:00. Room 2.2.D08

# Speaker: Yanely Zaldívar Gerpe. Universidad Carlos III de Madrid.

Title: Higher order recurrences and row sequences of Hermite-Padé approximation

Abstract: We obtain extensions of the Poincaré and Perron theorems for higher order recurrence relations and apply them to obtain an inverse type theorem for row sequences of (type II) Hermite-Padé approximation of a vector of formal power series.

# Title:  On several extremal problems in graph theory involving Gromov Hyperbolicity Constant

Abstract: If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\d$-\emph{hyperbolic} $($in the Gromov sense$)$ if any side of $T$ is contained in the $\d$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. We denote by $\d(X)$ the sharp hyperbolicity constant of $X$, i.e., $\d(X):=\inf\{\d\ge 0: \, X \, \text{ is$\d$-hyperbolic}\,\}\,.$

In this work we focus on the study of hyperbolic graphs. In general, to compute $\d(G)$ for a given graph $G$ is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Our main aim is to estimate the hyperbolicity constant of graphs belonging to particular families; more specifically, we obtain good upper and lower bounds for $\d(G)$ in terms of the order, size, diameter, girth, circumference, minimum and maximum degree of a graph.

# Title: Self-interlacing and stable orthogonal polynomials. Theory, examples and open problems

Abstract: Some non-symmetric tridiagonal matrices appear to be useful in some engineering questions,  for example, in network synthesis, stability theory etc. I will talk about orthogonal polynomials related to such tridiagonal matrices. Bessel polynomials will be presented as an example of the theory.

# Title: Asymptotic behavior of eigenvalues of a differential operator for discrete Gegenbauer-Sobolev orthonormal polynomials.

Abstract: In this talk, we consider discrete Sobolev inner products involving the Gegenbauer weight. The sequence of orthonormal polynomials with respect to this inner product are eigenfunctions of a differential operator. We establish the asymptotic behavior of the corresponding eigenvalues. Finally, we deduce the Mehler-Heine type asymptotics for the sequence of orthonormal polynomials and the location of the zeros of these polynomials.

# Title: An extension of Markov's theorem for mixed type Hermite-Padé approximants.

Abstract: Since their introduction in by Ch. Hermite, Hermite-Pad e approximants have been an invaluable tool in number theory, recently they have appeared in models coming from random matrix theory, biorthogonal ensembles and in some integrable systems. Due to classical results of A.A. Markov, the Cauchy transform of one measure plays a central role in the convergence theory of diagonal Pad e approximation. For type I and type II Hermite Pad e approximants, a similar place is occupied by the so-called Nikishin systems of functions, in particular, for such systems Markov type theorems are obtained. This talk deals with an extension of the Markov´s theorem for a kind of mixed type Hermite-Pad e approximants with respect to a Nikishin system of measures.

# Title:  Jacobi matrices on graph trees and multiple orthogonal polynomials

Thursday, October 26,  2017, 16:00. Room 2.2.D08

# Title: On Semiclassical Families of Bivariate Orthogonal Polynomials

Abstract: Orthogonal polynomials in two variables are studied as the natural generalization of orthogonal polynomials in one variable. Nevertheless, much work needs to be done in order to consider the theory of bivariate orthogonal polynomials as complete as the univariate case. In this work, we focus on extending well-known properties of univariate semiclassical orthogonal polynomials to the bivariate case. We also study algebraic and differential properties of a class of orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Finally, we extend the definition of coherent pairs of quasi-definite moment functionals to the bivariate case and deduce some of its consequences.

# Title: Funciones internas en el Espacio de Dirichlet

## Abstract: This thesis is focused on the study  of  matrix transformations of  matrices with  linear  functionals as entries. In particular, we study the Christoffel, Geronimus, and Geronimus-Uvarov transformations,  as well as,  their relation  with scalar orthogonal polynomials satisfying a higher order recurrence relation.

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