Workshop C7 - Special Functions and Orthogonal Polynomials


Organizers: Francisco Marcellán (Universidad Carlos III de Madrid & Instituto de Ciencias Matemáticas, Spain) - Kerstin Jordaan (University of Pretoria, South Africa) - Andrei Martinez-Finkelshtein (Universidad de Almería, Spain)

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Talks


July 17, 14:30 ~ 15:00 - Room B5

Orthogonality and approximation in Sobolev Spaces

Yuan Xu

University of Oregon, USA   -   yuan@uoregon.edu

The best approximation polynomials in a $L^2$ space are the partial sums of the Fourier orthogonal expansions in the same space. This can be extended to a Sobolev space, for which the orthogonality is defined with respect to an inner product that contains derivatives and approximation holds for functions and their derivatives simultaneously. We explain recent results in this talk, starting with approximation via Sobolev orthogonal polynomials on an interval with the Jacobi weight and continuing to results on the unit ball and on a triangle.

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July 17, 15:00 ~ 15:30 - Room B5

Matrix-valued orthogonal polynomials in several variables related to $\mathrm{SU}(n+1)\times \mathrm{SU}(n+1)$

Pablo Román

Universidad Nacional de Córdoba, Argentina   -   roman@famaf.unc.edu.ar

We study matrix-valued spherical functions for the symmetric pair $G = \mathrm{SU}(n+1)\times \mathrm{SU}(n+1)$ and $K = \mathrm{SU}(n+1)$ diagonally embedded. Under certain assumptions these functions give rise to a family of matrix-valued polynomials in several variables. These polynomials are orthogonal with respect to a matrix weight which is described explicitly and is irreducible, i.e. it does not have non-trivial invariants subspaces. From the group theoretic interpretation, we obtain two commuting matrix-valued differential operators having the matrix-valued orthogonal polynomials as eigenfunctions. Remarkably one of these differential operators is of order one. In the case $n=2$, we obtain polynomials in two-variables. The weight matrix in this case is supported on the interior of the Steiner hypocycloid.

Joint work with Erik Koelink (Radboud Universiteit) and Maarten van Pruijssen (Universität Paderborn).

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July 17, 15:30 ~ 16:00 - Room B5

Positive and negative results about integral representations for multivariable Bessel functions

Margit Rösler

Paderborn University, Germany   -   roesler@math.upb.de

There exist various interesting classes of multivariable Bessel functions, such as Bessel functions of matrix argument which are important in multivariate statistics, or the Bessel functions associated with root systems in Dunkl theory.

In this talk, we shall focus on integral representations for such multivariable Bessel functions which generalize the classical Sonine integral for the one-variable Bessel function $\, j_\alpha(z) = \, _0F_1(\alpha + 1; -z^2/4)$, $$ j_{\alpha+\beta}(z) = \frac{2\Gamma(\alpha + \beta +1)}{\Gamma(\alpha+1)\Gamma(\beta)}\int_0^1 j_\alpha(zt)\, t^{2\alpha+1}(1-t^2)^{\beta -1} dt \quad (\alpha > -1, \beta >0).$$ For Bessel functions of matrix argument there are analogous representations, going back already to Herz, and for certain Bessel functions associated with root systems of type $B$ there are similar results by Macdonald. In these known cases, however, the range of indices is restricted. Similar to the theory of Gindikin for Riesz measures, we shall extend these integral representations to larger index ranges in a distributional sense, und study under which conditions the representing distributions are actually given by positive measures. There turn out to be gaps in the admissible range of indices which are determined by the so-called Wallach set.

As a consequence, we shall obtain examples where the Dunkl intertwining operator between Dunkl operators associated with multiplicities $k\geq 0 $ and $k^\prime \geq k$ is not positive, which disproves a long-standing conjecture.

Joint work with Michael Voit (TU Dortmund, Germany).

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July 17, 16:00 ~ 16:30 - Room B5

A symbolic approach to perturbed second degree forms

Zélia da Rocha

Department of Mathematic of Faculty of Sciences of University of Porto, Portugal   -   mrdioh@fc.up.pt

The so-called second degree forms constitute an important set of regular forms that include as particular cases the four Chebyshev forms [1,2]. The second degree character of a form is preserved by several transformations among which is the perturbation that corresponds to a finite modification of the coefficients that appear in the recurrence relation of order two satisfied by the sequence of polynomials orthogonal with respect to that form. Thus, perturbed second degree forms are of second degree; furthermore, they are semi-classical. In this work, we present a symbolic algorithm [1] that allows to explicit several semi-classical properties of perturbed second degree forms, namely: the functional equation, the class of the form, the Stieltjes equation, a closed formula for the Stieltjes function, a structure relation and the second order linear differential equation. We apply the algorithm to the Chebyshev form of second kind and we explicit these properties for perturbations of several orders [1,2]. From these results, we can easily derive similar ones for the other three forms of Chebyshev [2].

Key words: Perturbed orthogonal polynomials; second-degree forms; semi-classical forms; Chebyshev forms; differential equations; symbolic computations.

[1] Z. da Rocha, A general method for deriving some semi-classical properties of perturbed second degree forms: the case of the Chebyshev form of second kind, J. Comput. Appl. Math., 296 (2016) 677-689.

[2] Z. da Rocha, On the second order differential equation satisfied by perturbed Chebyshev polynomials, J. Math. Anal., 7(1) (2016) 53-69.

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July 17, 17:00 ~ 17:30 - Room B5

Asymptotics of Chebyshev polynomials

Jacob Stordal Christiansen

Lund University, Sweden   -   stordal@maths.lth.se

Given an infinite compact set $\mathsf{E}\subset\mathbb{R}$, the $n$th Chebyshev polynomial, $T_n(z)$, is the unique monic polynomial of degree $n$ that minimizes the sup-norm $\Vert T_n\Vert_{\mathsf{E}}=:t_n$ on $\mathsf{E}$. While the lower bound $t_n\geq C(\mathsf{E})^n$ is classical, I shall briefly discuss upper bounds of the form \[ t_n\leq K\cdot C(\mathsf{E})^n \] for some $K>0$. Here, $C(\mathsf{E})$ is the logarithmic capacity of $\mathsf{E}$. The main focus of the talk will be on asymptotics of $T_n(z)$ (and $t_n$). I'll explain how to solve a 45+ year old conjecture of Widom for finite gap subsets of $\mathbb{R}$ and then discuss how one can go far beyond this simple class of subsets.

Joint work with Barry Simon (Caltech, Pasadena, USA), Peter Yuditskii (JKU, Linz, Austria) and Maxim Zinchenko (UNM, Albuquerque, USA).

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July 17, 17:30 ~ 18:00 - Room B5

Recurrence coefficients of orthogonal polynomials and the Painleve equations

Galina Filipuk

University of Warsaw, Poland   -   filipuk@mimuw.edu.pl

In this talk I shall review recent results on the connection of recurrence coefficients of orthogonal polynomials to the solutions of the Painleve equations. In particular, for certain weights the coefficients in the three term recurrence relation satisfy some forms of discrete Painleve equations, namely, dPI and dPIV. Moreover, I shall explain how to derive differential equations with respect to certain parameters.

The talk is based on the following two papers.

G. Filipuk, M. N. Rebocho, Differential equations for families of semi-classical orthogonal polynomials of class one, submitted.

G. Filipuk, M. N. Rebocho, Discrete Painleve equations for recurrence coeffcients of Laguerre-Hahn orthogonal polynomials of class one, Integral Transforms and Special Functions 27 (2016), 548--565.

Joint work with M.N Rebocho (UBI, Portugal).

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July 17, 18:00 ~ 18:30 - Room B5

On three-fold symmetric polynomials with a classical behaviour

Ana F. Loureiro

University of Kent, U.K.   -   A.Loureiro@kent.ac.uk

I will discuss sequences of polynomials of a single variable that are orthogonal with respect to a vector of weights defined in the complex plane. Such polynomial sequences satisfy a recurrence relation of finite (and fixed) order higher than 2. The main focus will be on polynomial sequences possessing a three-fold symmetry and whose multiple orthogonality is preserved under the action of the derivative operator.

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July 17, 18:30 ~ 19:00 - Room B5

Bergman Polynomials and Torsional Rigidty

Brian Simanek

Baylor University, United States of America   -   Brian_Simanek@Baylor.edu

The torsional rigidity of a simply connected domain is a constant indicative of the resistance to twisting of a cylindrical beam with the given cross-section. We will show how one can use Bergman polynomials to calculate or estimate torsional rigidity in very general cases. A special emphasis will be placed on the simplicity of the necessary calculations and several examples that display the efficiency of our algorithm.

Joint work with Matthew Fleeman (Baylor University).

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July 18, 14:30 ~ 15:00 - Room B5

Zeros of Wronskians of Hermite polynomials

Walter van Assche

KU Leuven, Belgium   -   walter.vanassche@kuleuven.be

Wronskians of Hermite polynomials appear as exceptional Hermite polynomials but also in rational solutions of Painlevé IV, where the rational solutions are in terms of generalized Hermite polynomials and generalized Okamoto polynomials. For exceptional Hermite polynomials it is important to know the zeros of such Wronskians, because real zeros are not allowed in the construction of exceptional Hermite polynomials. For the rational solutions of Painlevé IV the zeros give information of the poles of Painlevé transcendents. We will discuss recent results of Felder, Hemery and Veselov (2012) and Buckingham (2017) about the location of the zeros of these Wronskians.

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July 18, 15:00 ~ 15:30 - Room B5

The Kontsevich matrix integral and Painlevé hierarchy

Marco Bertola

SISSA (Trieste, Italy) and Concordia University (Montreal, Canada), Italy/Canada   -   Marco.Bertola@sissa.it

The Kontsevich integral is a matrix integral (aka "Matrix Airy function") whose logarithm, in the appropriate formal limit, generates the intersection numbers on $\mathcal M_{g,n}$. In the same formal limit it is also a particular tau function of the KdV hierarchy; truncation of the times yields thus tau functions of the first Painlevé hierarchy. This, however is a purely formal manipulation that pays no attention to issues of convergence.

The talk will try to address two issues: Issue 1: how to make an analytic sense of the convergence of the Kontsevich integral to a tau function for a member of the Painlevé I hierarchy? Which particular solution(s) does it converge to? Where (for which range of the parameters)?

Issue 2: it is known that (in fact for any $\beta$) the correlation functions of K points in the $GUE_\beta$ ensemble of size N are dual to the correlation functions of N points in the $GUE_{4/\beta}$ of size $K$. For $\beta=2$ they are self-dual.

Consider $\beta=2$: this duality is lost if the matrix model is not Gaussian; however we show that the duality resurfaces in the scaling limit near the edge (soft and hard) of the spectrum.

In particular we want to show that the correlation functions of $K$ points near the edge of the spectrum converge to the Kontsevich integral of size $K$ as $N\to \infty$.

This line of reasoning was used by Okounkov in the GUE$_2$ for his "edge of the spectrum model". This is based on joint work with Mattia Cafasso (Angers).

Time permitting I will discuss a work-in-progress with G. Ruzza (SISSA) extending these results to the generating function of open intersection numbers.

Joint work with Mattia Cafasso, University of Angers, France and Giulio Ruzza, SISSA, Italy..

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July 18, 15:30 ~ 16:00 - Room B5

Six-Vertex Model, Yang—Baxter Equations, and Orthogonal Polynomials

Pavel Bleher

Indiana University—Purdue University Indianapolis, USA   -   pbleher@iupui.edu

We will discuss an exact solution of the six-vertex model with domain wall and half-turn boundary conditions in each of the phase regions, disordered, ferroelectric, and anti-ferroelectric. The solution is based on the Yang—Baxter equations, the Izergin—Korepin—Kuperberg determinantal formulae, the Zinn-Justin transform, and on the Riemann—Hilbert approach to the asymptotic analysis of orthogonal polynomials, of both continuous and discrete type. This is a joint work with Karl Liechty.

Joint work with Karl Liechty, DePaul University, Chicago, U.S.A..

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July 18, 16:00 ~ 16:30 - Room B5

The normal matrix model, multiple orthogonal polynomials and the mother body problem

Guilherme Silva

University of Michigan, USA   -   silvag@umich.edu

The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth.

In this presentation we consider the normal matrix model with cubic plus linear potential. Following previous works of Bleher & Kuijlaars and Kuijlaars & López, we introduce multiple orthogonal polynomials that should, in the large degree limit, coincide with the average characteristic polynomial of the normal matrix model. Developing the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem we obtain asymptotics for these polynomials, in particular determining their limiting zero distribution $\mu_*$, which will be the main topic of our talk.

We show that the measure $\mu_*$ can be characterized through the mother body problem associated with the eigenvalues of the normal matrix model. Interestingly, we also find a phase transition for the measure $\mu_*$, showing that its support undergoes a ''one-cut to three-cut'' phase transition.

At the technical level, the construction of $\mu_*$ involves deformation techniques for quadratic differentials that we also plan to present (in case time permits).

Joint work with Pavel Bleher (IUPUI, USA).

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July 18, 17:00 ~ 17:50 - Room B5

Painleve' equations and universality results in Hamiltonian PDEs

Tamara Grava

SISSA/BRISTOL, ITALY/UK   -   grava@sissa.it

We describe the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as a perturbation of elliptic or hyperbolic systems of hydrodynamic type with two components. Near the points of singularity formation for the unperturbed system, the solutions of the perturbed systems are described by particular solutions to the Painleve' equations. Painleve' equations play in this case the nonlinear analogue of the Airy function and Pearcy integral that are used to describe the semiclassical limit of the linear Schroedinger equation near transition regimes. The important feature of this result is that generic solutions to Hamiltonian perturbations of hyperbolic or elliptic systems behave, near critical points, as solutions to an integrable equation.

Joint work with B. Dubrovin (SISSA), C. Klein (Dijon), A. Moro (New Castle).

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July 18, 18:00 ~ 18:30 - Room B5

On difference equations for orthogonal polynomials on nonuniform lattices

Mama Foupouagnigni

University of Yaounde I and African Institute for Mathematical Sciences, Cameroon   -   foupouagnigni@gmail.com

Classical orthogonal polynomials on nonuniform lattices $x(s)$ are known to satisfy a second-order divided difference equation \[ \phi(x(s))\,D_x^2P_n(x(s))+\psi(x(s))\,S_x\,D_xP_n(x(s))+\lambda_n\,P_n(x(s))=0, \] with the divided-difference operators $D_x$ and $S_x$ defined by \[ D_xf(x(s))={f(x(s+{1\over 2})-f(x(s-{1\over 2})\over x(s+{1\over 2})-x(s-{1\over 2})}, \;\;S_xf(x(s))={f(x(s+{1\over 2})+f(x(s-{1\over 2})\over 2}, \] where $\phi$ and $\psi$ are polynomials of degree maximum 2 and 1 respectively and $\lambda_n$ is a constant term with respect to $x(s)$. The lattice $x(s)$ is defined as \[ x(s)=\left\{\begin{array}{lll} c_1\,q^{s}+c_2\,q^{-s}+c_3&\;\;if& q\neq 1\\ c_4\,s^2+c_5\,s +c_6& \;\;if& q=1. \end{array} \right. \]

We first derive an appropriate polynomial basis for the operators $D_x$ and $S_x$, denoted $(F_n)_n$ and fulfilling \[ D_x F_n=a_n\,F_{n-1},\;\;S_x F_n=b_b\,F_n+c_n\,F_{n-1}, \] where $a_n,\,b_n,\,$ and $c_n$ are given constants.

Secondly, we use this basis combined with properties of the operators $D_x$ and $S_x$ to derive many properties of classical orthogonal polynomials on nonuniform lattice including: 1) Proof of existence of polynomial solution of the above mentioned second-order divided difference equation; 2) providing algorithmic method to find explicit solution to holonomic divided-difference equation and 3) the derivation of some structure relations for classical orthogonal polynomials on nonuniform lattices such the determination of the coefficients of the connection and the linearisation problems involving classical orthogonal polynomials on nonuniform lattices.

We end up by showing how this basis could be used, together with the operators operators $D_x$ and $S_x$ to derive a fourth-order partial divided-difference equations for bivariate classical orthogonal polynomials on the nonuniform lattice.

Joint work with Wolfram Koepf (University of Kassel, Germany), Salifou Mboutngam (University of Maroua, Cameroon), Maurice Kenfack-Nangho (University of Dschang, Cameroon), Daniel Duviol Tcheutia (University of Kassel, Germany) and Patrick Njionou Sadjang (University of Douala, Cameroon).

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July 18, 18:30 ~ 19:00 - Room B5

Tridiagonalization of the Heun equation

Luc Vinet

Université de Montréal, Canada   -   luc.vinet@umontreal.ca

It will be explained that the tridiagonalization of the hypergeometric operator $L$ yields the generic Heun operator $M$. The algebra generated by the operators $L$, $M$ and $ Z = [L,M]$ will be seen to be quadratic and to form a one-parameter generalization of the Racah algebra. Racah-Heun orthogonal polynomials will be introduced as overlap coefficients between the eigenfunctions of the operators $L$ and $M$. An interpretation in terms of the Racah problem for the $su(1,1)$ algebra and separation of variables in a superintegrable system will be discussed.

Joint work with F. Alberto Grünbaum (UC Berkeley, USA) and Alexei Zhedanov (Renmin U., China).

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July 19, 14:30 ~ 15:20 - Room B5

Ahlfors problem for polynomials

Peter Yuditskii

Johannes Kepler University of Linz, Austria   -   peter.yuditskii@gmail.com

We raise a conjecture that asymptotics for Chebyshev polynomials in a complex domain can be given in terms of the reproducing kernels of a suitable Hilbert space of analytic functions in this domain. It is based on two classical results due to Garabedian and Widom. To support this conjecture we study asymptotics for Ahlfors extremal polynomials in the complement to a system of intervals on $\mathbb{R}$, arcs on $\mathbb{T}$, and its continuous counterpart.

Joint work with Benjamin Eichinger (Johannes Kepler University of Linz).

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July 19, 15:30 ~ 16:00 - Room B5

Construction of optimal set of two quadrature rules in the sense of Borges

Marija Stani\'c

University of Kragujevac, Faculty of Science, Department of Mathematics and Informatics, Serbia   -   stanicm@kg.ac.rs

In this paper we investigate the numerical method for construction of optimal set of quadrature rules in the sense of Borges [Numer. Math. {\bf67} (1994), 271--288] for two definite integrals with the same integrand and interval of integration, but with different weight functions, related to an arbitrary multi--index. Presented method is illustrated by numerical example.

Joint work with Tatjana Tomovi\'c (University of Kragujevac, Faculty of Science, Department of Mathematics and Informatics, Serbia).

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July 19, 16:00 ~ 16:30 - Room B5

Outlier detection through Christoffel functions

Bernhard Beckermann

University of Lille, France   -   bbecker@math.univ-lille1.fr

We discuss how recent results on asymptotics for the Christoffel kernel built with orthogonal polynomials of one or several complex variables can help to detect outliers in a cloud of points.

Joint work with Mihai Putinar (University of California at Santa Barbara), Edward B. Saff (Vanderbilt University) and Nikos Stylianopoulos (University of Cyprus).

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July 19, 17:00 ~ 17:30 - Room B5

Self-interlacing and Hurwitz stable polynomials in applications to orthogonal polynomials and integrable systems

Mikhail Tyaglov

Shanghai Jiao Tong University, China   -   tyaglov@sjtu.edu.cn

In the talk, we describe some interrelations between theory of orthogonal polynomials, moment theory, and the root distributions of polynomials. The recent discovered self-interlacing polynomials and their dual Hurwitz stable polynomials play an important role in these interrelations. Possible applications of Hurwitz stable and self-interlacing orthogonal polynomials to integrable systems and quantum transportation are discussed.

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July 19, 17:30 ~ 18:00 - Room B5

A connection between orthogonal polynomials on the unit circle and the real line, via CMV matrices

María-José Cantero

University of Zaragoza, Spain   -   mjcante@unizar.es

It is very well known the connection between orthogonal polynomials on the unit circle and orthogonal polynomials on the real line, given by Szegö. This connection, based on a map which transforms the unit circle onto certain interval of the real line, induces a one-to-one correspondence between symmetric measures on the unit circle and measures on the mentioned interval.

A new relation between these two kind of polynomials has been recently discovered by Derevyagin, Vinet and Zhedanov (DVZ). The DVZ connection starts from a factorization of real CMV matrices ${\cal C}$ (unitary analogue of Jacobi matrices) into two tridiagonal factors, whose sum yields a Jacobi matrix ${\cal K}$. The authors provide closed formulas for the orthogonality measure and the orthogonal polynomials related to ${\cal K}$ in terms of those corresponding to ${\cal C}$ . DVZ consider even a more ambitious problem in which the Jacobi matrix $\cal K$ is a linear combination of the referred CMV factors (generalized DVZ connection). Regarding this generalization, their main result refers to the Jacobi matrix $\cal K$ built out of the Jacobi polynomials on the unit circle. They obtain a connection between the orthogonal polynomials of $\cal K$ and the so called big $-1$ Jacobi polynomials. However, for generalized DVZ, the relation between the orthogonal polynomials and orthogonality measures of $\cal K$ and $\cal C$ is missing.

We will present a different approach to this connection which allows us to go further than DVZ. We start by using CMV tools to obtain directly the orthogonal polynomials associated with ${\cal K}$ in terms of the basis related to ${\cal C}$, and then we use this to discover the relation between the corresponding orthogonality measures. The advantages of our approach are more evident for the generalized DVZ connection, where we obtain explicit formulas for the relation between the orthogonal polynomials and orthogonality measures associated with $\cal K$ and $\cal C$. The utility of these results will be illustrated with some examples providing new families of orthogonal polynomials on the real line.

Joint work with Francisco Marcellán (University Carlos III of Madrid, Spain), Leandro Moral (University of Zaragoza, Spain) and Luis Velázquez (University of Zaragoza, Spain).

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July 19, 18:00 ~ 18:30 - Room B5

Christoffel formula for Christoffel-Darboux kernels on the unit circle

Alagacone Sri Ranga

UNESP - Universidade Estadual Paulista , Brazil   -   ranga@ibilce.unesp.br

Given a nontrivial positive measure $\mu$ on the unit circle, the associated Christoffel-Darboux kernels can be considered as polynomials in $z$ given by $K_n(z, w;\mu) = \sum_{k=0}^{n}\overline{\varphi_{k}(w;\mu)}\,\varphi_{k}(z;\mu)$, $n \geq 0$, where $\varphi_{k}(\cdot; \mu)$ are the orthonormal polynomials with respect to the measure $\mu$. Let the positive measure $\nu$ on the unit circle be given by $d \nu(z) = z^{-m}G_{2m}(z)\, d \mu(z)$, where $G_{2m}$ is a conjugate reciprocal polynomial of exact degree $2m$ such that $z^{-m}G_{2m}(z)$ is non-negative within the support of the $\mu$. We consider the determinantal formula expressing $\{K_n(z,w;\nu)\}_{n \geq 0}$ directly in terms of $\{K_n(z,w;\mu)\}_{n \geq 0}$. Special attention is given to the case $w =1$. Example are also given.

Joint work with Cleonice F. Bracciali (Universidade Estadual Paulista, Brazil), Andrei Martinez-Finkelshtein (Universidad de Almeria, Spain) and Daniel O. Veronese (Universidade Federal do Triangulo Mineiro, Brazil).

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July 19, 18:30 ~ 19:00 - Room B5

The projective ensemble and distribution of points in odd-dimensional spheres

Ujué Etayo

Universidad de Cantabria, Spain   -   etayomu@unican.es

We define a determinantal point process on the complex projective space that reduces to the so-called spherical ensemble for complex dimension 1 under identification of the 2-sphere with the Riemann sphere.

Through this determinantal point process we propose a point processs in odd-dimensional spheres that produces fairly well-distributed points, in the sense that the expected value of the Riesz 2-energy for these collections of points is smaller than all previously known bounds.

Joint work with Carlos Beltrán (Universidad de Cantabria).

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Posters


Galoisian Approach to Orthogonal Polynomials

Primitivo Acosta-Humánez

Universidad Simón Bolívar, Barranquilla, Colombia   -   primitivo.acosta@unisimonbolivar.edu.co

In this poster I will present some links between Classical and Differential Galois Theory with the Theory of Orthogonal Polynomials. The starting point corresponds to some general results about the Galois groups of orthogonal polynomials and differential equations with orthogonal polynomials as solutions. In particular, we compute the Galois group of some families of classical polynomials with base field the rational numbers as well the differential Galois groups of some families of differential equations, with base differential field the rational functions, involving orthogonal polynomials in their solutions and spectra.

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Collocation spline interpolation for numerical solution of high-order Lidstone-type Boundary Value Problems

Maria Italia Gualtieri, Anna Napoli

University of Calabria, Italy   -   gualtieri@mat.unical.it, anna.napoli@unical.it

In this paper we consider Lidstone and Complementary Lidstone expansions for a sufficiently smooth function ([1]). The corresponding interpolating polynomials have been used in [2,3,4] in the field of interpolation theory and its application, particularly, for the numerical solution of boundary value problems.

Here we present a new collocation method for solving nonlinear boundary value problems of high-order $m$ with Lidstone and Complementary Lidstone boundary conditions. The exact solution is approximated by spline interpolation functions of order $m+2$ which are written in the basis of Lidstone and Complementary Lidstone polynomials, respectively, for even and odd-order problems. The use of piecewise polynomials and spline-type functions allows to overcome the drawback of polynomial interpolation which may produce widely oscillatory approximations.

Some results on local and global errors are given. The cases of third and fourth order boundary value problems have been analyzed in detail and some numerical tests are given in order to confirm the validity of the presented method.

[1] G.J. Lidstone, Notes on the extension of Aitken's theorem (for polynomial interpolation) to the Everett types, Proc. Edinburgh Math. Soc. 2 (1929), 16-19.

[2] R.P. Agarwal and P.J.Y. Wong, Lidstone polynomials and boundary value problems, Comput. Math. Appl., 17(1989), 1397-1421.

[3] F.A. Costabile, F. Dell'Accio and R. Luceri, Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values, J. Comp. Appl. Math., 175 (2005), 77-99.

[4] R.P. Agarwal, S. Pinelas and P.J.Y. Wong, Complementary Lidstone interpolation and boundary value problems, J. Ineq. Appl., 2009(1) (2009), 1-30.

Joint work with Francesco Aldo Costabile (University of Calabria, Italy).

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On supersymmetric eigenvectors of the $5D$ discrete Fourier transform

Natig Atakishiyev

Instituto de Matemáticas, UNAM, Unidad Cuernavaca, México   -   natig@matcuer.unam.mx

An explicit form of a discrete analogue of the quantum number operator, constructed in terms of the difference lowering and raising operators that govern eigenvectors of the 5D discrete (finite) Fourier transform ${\Phi}^{(5)}$, has been explored. This discrete number operator ${\bf \cal N}^{(5)}$ has distinct eigenvalues, which are employed to systematically classify eigenvectors of the ${\Phi}^{(5)}$, thus avoiding the ambiguity caused by the well-known degeneracy of the eigenvalues of the latter operator. In addition, we show that the hidden symmetry of the discrete number operator ${\bf \cal N}^{(5)}$ manifests itself in the form of the unitary Lie superalgebra $psl(5|5)$.

Joint work with Mesuma Atakishiyeva (Universidad Autónoma del Estado de Morelos, México).

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On perturbations of $g$-fractions and related consequences

Kiran Kumar Behera

Indian Institute of Technology Roorkee, India   -   krn.behera@gmail.com

The purpose of the work is to investigate some structural and qualitative aspects of two different perturbations of the parameters of $g$-fractions. In this context, the concept of gap-$g$-fractions is introduced. While tail sequences of continued fractions play a significant role in the first perturbation, Schur fractions are used in the second perturbation of the $g$-parameters that are considered. Illustrations are provided using Gaussian hypergeometric functions.

Joint work with A. Swaminathan (Indian Institute of Technology Roorkee).

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Hankel determinants and special function solutions of Painlevé IV

Alfredo Deaño

SMSAS, University of Kent, United Kingdom   -   A.Deano-Cabrera@kent.ac.uk

We consider $n\times n$ Hankel determinants involving moments of discontinuous Hermite weights defined on the real line, and we present asymptotic results as $n\to\infty$. These Hankel determinants are related to Laguerre and Hermite semiclassical polynomials, to special function solutions of the Painlevé IV equation, which are constructed using Wronskians of parabolic cylinder functions, and also to thinning processes of Gaussian ensembles of random matrices.

Joint work with Christophe Charlier (Université Catholique de Louvain, Belgium).

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Fourth order partial differential equations for Krall-type orthogonal polynomials on the triangle

Lidia Fernández

Universidad de Granada, Spain   -   lidiafr@ugr.es

We construct bivariate polynomials orthogonal with respect to a Krall-type inner product on the triangle defined by adding Krall terms over the border and the vertexes to the classical inner product. We prove that these Krall-type orthogonal polynomials satisfy fourth order partial differential equations with polynomial coefficients, as an extension of the classical theory introduced by H. L. Krall in the 40's. Also, a connection formula between classical orthogonal polynomials on the bidimensional simplex and orthogonal polynomials associated with Krall-type inner product is deduced.

Joint work with Antonia M. Delgado and Teresa E. Pérez (Universidad de Granada, Spain).

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Toeplitz minors for Szegö and Fisher-Hartwig symbols

David García-García

Universidade de Lisboa, Portugal   -   dgarciagarcia@fc.ul.pt

We study minors of Toeplitz matrices of both finite and large dimension, comprising also the case of symbols with Fisher-Hartwig singularities. We express the minors in terms of specializations of symmetric polynomials. Several implications of this formulation are presented, including explicit formulas for a Selberg-Morris integral with two Schur polynomials and for the specialization of skew-Schur polynomials. For the latter result, the inverse of a Toeplitz matrix with a pure Fisher-Hartwig singularity is computed, using both our results on minors and the Duduchava-Roch formula.

References:

[1] D. Bump and P. Diaconis, “Toeplitz minors”, J. Comb. Th. (A) 97 (2002).

[2] D. García-García and M. Tierz, “Toeplitz minors for Szeg? and Fisher-Hartwig symbols”, preprint [arXiv:1706.02574].

[3] A. Böttcher, “The Duduchava–Roch Formula”, in Recent Trends in Operator Theory and PDEs - The Roland Duduchava Anniversary Volume (2017).

Joint work with Miguel Tierz (Universidade de Lisboa, Portugal).

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Generalized Freud polynomials and the Painleve equations

Kerstin Jordaan

University of South Africa   -   jordakh@unisa.ac.za

We investigate certain properties of monic polynomials orthogonal with respect to a semi-classical generalized Freud weight $$ w_{\lambda}(x;t)=|x|^{2\lambda +1}\exp\left(-x^4+tx^2\right), $$ where $x \in \mathbb{R}$, $\lambda>0$ and $t \in \mathbb{R}$. Generalized Freud polynomials arise from a symmetrization of semi-classical Laguerre polynomials. We prove that the coefficients in the three-term recurrence relation associated with the generalized Freud weight $w_{\lambda}(x;t)$ can be expressed in terms of Wronskians of parabolic cylinder functions that appear in the description of special function solutions of the fourth Painleve equation. This closed form expression for the recurrence coefficients allows the investigation of certain properties of the generalized Freud polynomials. We obtain an explicit formulation for the generalized Freud polynomials in terms of the recurrence coefficients, investigate the higher order moments as well as the Pearson equation satisfied by the generalized Freud weight $w_{\lambda}(x;t)$. We also derive a differential-difference equation using a method due to Shohat and second-order linear ordinary differential equation satisfied by the polynomials. Further, we provide an extension of Freud's conjecture for the recurrence coefficient $\beta_n(t;\lambda)$ associated with the generalized Freud weight.

Joint work with Peter Clarkson (University of Kent, UK) and Abey Kelil (University of Pretoria, South Africa).

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Eigenvalues of a differential operator for a family of Sobolev orthogonal polynomials

Juan Francisco Mañas-Mañas

Universidad de Almería, España   -   jmm939@ual.es

We consider the Sobolev inner product \[ (f,g)_S:=\int_{-1}^1f(x)g(x)(1-x^2)^{\alpha}dx+M\big[f^{(j)}(-1)g^{(j)}(-1)+f^{(j)}(1)g^{(j)}(1)\big], \] where $\alpha>-1,$ $j\geq0$ and $M>0.$

We denote by $\{B_n^{(\alpha,j)}\}_{n\geq0}$ the sequence of orthonormal polynomials with respect to this inner product. These polynomials are called Gegenbauer-Sobolev orthogonal polynomials since involved the classical Gegenbauer weight.

In [1] the authors give conditions to assure that there is a operator differential equation, $\mathbf{L}$, such that the polynomials $B_n^{(\alpha,j)}$ are eigenfunctions of

\[\mathbf{L}B_n^{(\alpha,j)}(x)=\lambda_nB_n^{(\alpha,j)}(x),\] where $\lambda_n$ are the corresponding eigenvalues of $\mathbf{L}.$

For certain applications we are interested in properties of the related polynomial kernel scaled by the eigenvalues: \[K^r(x,y)= \sum_{i=0}^{\infty}\lambda_i^{-r} \frac{B_i^{(\alpha,j)}(x)B_i^{(\alpha,j)}(y)}{(B_i^{(\alpha,j)},B_i^{(\alpha,j)})_S}\]

If the series converges for some $r_0$, then $K^r$ is a reproducing kernel for all $r\geq r_0$. Therefore the \emph{minimal} $r_0>0$ for which the series converges for all $r > r_0$ is a number of interest.

Our main result is to determinate this $r_0$, it is shown in [2]

\[ \lim_{n\to+\infty}\frac{\log(\max_{x\in [-1,1]}|B_n^{(\alpha,j)}(x)|)}{\log(\lambda_n)}=r_{0}. \]

Finally, we study the Mehler-Heine type asymptotics for the polynomials $B_n^{(\alpha,j)}$.

[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.

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

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Zeros of some families of Sobolev orthogonal polynomials

Juan José Moreno-Balcázar

Universidad de Almería, Spain   -   balcazar@ual.es

The computation of the zeros of the Sobolev orthogonal polynomials (SOP) can be rather complicated. Thus, the study of the asymptotic behavior of these zeros can be useful as an approximation of them. In the literature the use of Mehler--Heine type formulae has permitted to obtain asymptotic approximations to the zeros of the SOP in terms of the zeros of Bessel functions or combination of these functions. From a general analytical result for discrete SOP, we will show some examples focusing our attention on the numerical experiments.

Joint work with Juan F. Mañas-Mañas (Universidad de Almería).

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Fast and accurate algorithm for the generalized exponential integral $E_{\nu}(x)$ for positive real order

Guillermo Navas-Palencia

Universitat Politècnica de Catalunya, Spain   -   g.navas.palencia@gmail.com

We describe an algorithm for the numerical evaluation of the generalized exponential integral $E_{\nu}(x)$ for positive values of $\nu$ and $x$. The generalized exponential integral appears in many fields of physics and engineering and plays an important role in some exponentially-improved asymptotic expansions for the confluent hypergeometric function. The performance and accuracy of the resulting algorithm is analysed and compared with open-source software packages. This analysis shows that our implementation is competitive and more robust than other state-of-the-art codes. The described algorithm is part of the author's library Chypergeo.

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Zeros of bivariate classical orthogonal polynomials on the unit disk

Teresa E. Perez

Universidad de Granada, Spain   -   tperez@ugr.es

The behaviour of the zeros of orthogonal polynomials in one variable has been studied extensively for years. In the positive-definite case, it is well know that the $n$th polynomial has exactly $n$ distinct zeros, that have important applications in quadrature formulae. In its classical sense, a zero of a bivariate orthogonal polynomial is an algebraic curve, and depends on the chosen basis in every case.

As far as we know, in two variables, the problem was tackled by Charles Hermite in 1865 for the biorthogonal basis of classical orthogonal polynomials on the unit disk. This work is devoted to show some old and new results, difficulties and open problems about this subject.

Joint work with Antonia M. Delgado (Universidad de Granada, Spain), Lidia Fernández (Universidad de Granada, Spain) and Miguel A. Piñar (Universidad de Granada, Spain).

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Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

David Puertas Centeno

Universidad de Granada, España   -   vidda@correo.ugr.es

The determination of the physical entropies (Rényi, Shannon, Tsallis) of high-dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control the wavefunctions of the stationary states. For the D-dimensional hydrogenic and oscillator-like systems, the wavefunctions of the corresponding bound states are controlled by the Laguerre $(\mathcal L^{(\alpha)}_m(x))$ and Gegenbauer $(\mathcal C^{(\alpha)}_m (x))$ polynomials in both position and momentum spaces, where the parameter $\alpha$ linearly depends on D. In this work we study the asymptotic behavior as $\alpha\to\infty$ of the associated entropy-like integral functionals of these two families of hypergeometric polynomials. Finally, further extension of these results are applied to hyperspherical harmonics together with some quantum applications.

Joint work with Nico M. Temme (IAA, Netherlands), Irene V. Toranzo (Universidad de Granada, Spain) and Jesús S. Dehesa (Universidad de Granada, Spain).

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A new framework for numerical analysis of nonlinear systems: the significance of the Stahl's theory and analytic continuation via Padé approximants

Sina Sadeghi Baghsorkhi

University of Michigan, United States   -   sinasb@umich.edu

An appropriate embedding of polynomial systems of equations in a Riemann surface renders the variables as functions of a single complex variable. The relatively recent developments in the theory of approximation of multivalued functions in the extended complex plane provides a set of powerful tools to solve and analyze this class of problems. The underlying concepts, namely the algebraic curves and the Stahl's theory are presented along with a critical application in the voltage collapse study of electricity networks.

Joint work with Nikolay Ikonomov (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences) and Sergey Suetin (Steklov Mathematical Institute, Russian Academy of Sciences).

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On Cooley-Tukey-type-algorithms based on generalized Chebyshev polynomials

Bastian Seifert

Ansbach University of Applied Sciences, Germany   -   bastian.seifert@hs-ansbach.de

Weyl groups associated to semisimple, compact Lie groups give rise to generalized Chebyshev polynomials. In recent years those of second kind have found a wide variety of applications, ranging from discretization of partial differential equations to cubature formulas on special domains. In this presentation we illustrate how one can use generalized Chebyshev polynomials of the first kind to develop Cooley-Tukey-type-algorithms in the setting of algebraic signal processing theory for the lattices they constitute. As an illustrating example, we show how one can use this technique to process voxel data on the face centered lattice.

Joint work with Knut Hüper (University of Würzburg, Germany) and Christian Uhl (Ansbach University of Applied Sciences, Germany).

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Linearization and Krein-like functionals of hypergeometric orthogonal polynomials by means of Lauricella functions

Irene Valero Toranzo

Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, Spain   -   ivtoranzo@ugr.es

The Krein-like functionals of the hypergeometric orthogonal polynomials $\{p_{n}(x) \}$ with kernel of the form $x^{s}[\omega(x)]^{\beta}p_{m_{1}}(x)\ldots p_{m_{r}}(x)$, being $\omega(x)$ the weight function on the interval $\Delta\in\mathbb{R}$, are considered and analytically computed by means of the Srivastava linearization method. The particular ($r=2$)-functionals of the form $x^{s}[\omega(x)]^{\beta}p_{n}(x)p_{m}(x)$, which are particularly relevant in quantum physics, are explicitly given in terms of the degrees and the characteristic parameters of the polynomials. They include the well-known power moments $\langle x^{s}\rangle_{n}$ and the novel Krein-like moments $\langle [\omega(x)]^{k}\rangle_{n}$, where $\langle f(x)\rangle_{n} :=\int_{\Delta} f(x)\rho_{n}(x)\, dx$ and $\rho_{n}(x) = \omega(x)p_{n}^{2}(x)$ is the Rakhmanov density of the polynomials. Moreover, various related types of exponential and logarithmic functionals of the form $\langle x^{k}e^{-\alpha x}\rangle_{n}$, $\langle (\log x)^{k}\rangle_{n}$ and $\langle [\omega(x)]^{k}\log \omega(x)\rangle_{n}$, are also investigated.

References

J. S Dehesa, J. J. Moreno-Balcázar, I. V. Toranzo, Linearization and Krein-like functionals of hypergeometric orthogonal polynomials by means of Lauricella functions. Preprint in preparation 2017.

P. Sánchez-Moreno, J. S. Dehesa, A. Zarzo, A. Guerrero, Rényi entropies, $L_{q}$ norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions, Applied Mathematics and Computation 223 (2013) 25–33.

H. M. Srivastava, A. W. Niukkanen, Some Clebsch-Gordan Type Linearization Relations and Associated Families of Dirichlet Integrals, Math. Comput. Model 37 (2003) 245-250.

E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, Math. USSR Sb. 32 (1977) 199-213.

M. Krein, N. I. Ahiezer, Some questions in the Theory of Moments (AMS, Providence, 1962).

F. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989).

Joint work with J. S. Dehesa (Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, Granada 18071, Spain; Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada 18071, Spain) and J. J. Moreno-Balcázar (Departamento de Matemáticas, Universidad de Almería, Almería 04120, Spain; Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada 18071, Spain).

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FoCM 2017, based on a nodethirtythree design.