ENG 
ESP Session S32 - Special functions and orthogonal polynomials
Talks
Monday, July 12, 11:00 ~ 12:00 UTC-3
Linear spectral transforms, matrix factorizations and orthogonal polynomials
Francisco Marcellán
Universidad Carlos III de Madrid, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $u$ be a quasi-definite linear functional defined on the linear space of polynomials $\mathbb{P}.$ For such a functional we can define a sequence of monic orthogonal polynomials (SMOP in short) $(P_n)_{n\geq 0},$ which satisfies a three term recurrence relation. Shifting one unity the recurrence coefficient indices we get the sequence of associated polynomials of the first kind$(P_n^{(1)})_{n\geq 0}$ which are orthogonal with respect to a linear functional denoted by $u^{(1)}$.
In the literature two special spectral transformations of the functional $u$ are studied: the canonical Christoffel transformation $\widetilde{u}=(x-c) u$ and the canonical Geronimus transformation $\widehat {u}= (x-c)^{-1} {u}+M\delta_c$ , where $c$ is a fixed complex number, $M $ is a free parameter and $\delta_c$ is the linear functional defined on $\mathbb{P}$ as $<\delta_{c},p(x) >=p(c).$ They constitue a generating system of the so called linear spectral transformation set analyzed in [2]. For the Christoffel transformation with SMOP $(\widetilde P_n)_{n\geq 0}$, we are interested in analyzing the relation between the linear functionals $ u^{(1)}$ and $\widetilde{u}^{(1)}.$ There, the super index denotes the linear functionals associated with the orthogonal polynomial sequences of the first kind $(P_n^{(1)})_{n\geq 0}$ and $(\widetilde P_n^{(1)})_{n\geq 0},$ respectively. This problem is also studied for Geronimus transformations. Here we give close relations between their corresponding monic Jacobi matrices by using the LU and UL factorizations.
For more information, see [1].
References
[1] J. C. García Ardila, F. Marcellán, P. H. Villamil-Hernández, \textit{Associated orthogonal polynomials of the first kind and Darboux transformations}, arXiv:2103.0232, [math.CA] 23 March 2021
[2] A. Zhedanov, \textit{Rational spectral transformations and orthogonal polynomials}, J. Comput. Appl. Math. \textbf{85} (1997), 67-86.
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Joint work with Juan Carlos Garcia Ardila (Universidad Politécnica de Madrid, España) and Paul H. Villamil-Hernández (Universidad Carlos III de Madrid, España).
Monday, July 12, 12:00 ~ 13:00 UTC-3
Some properties of the generalized mixed type Bernoulli-Gegenbauer polynomials
Yamilet Quintana
Simón Bolívar University, Venezuela - This email address is being protected from spambots. You need JavaScript enabled to view it.
The generalized mixed type Bernoulli-Gegenbauer polynomials of order $\alpha>-\frac{1}{2}$ are special polynomials obtained by use of the generating function method. These polynomials represent an interesting mixture between two classes of special functions, namely generalized Bernoulli polynomials and Gegenbauer polynomials. In this talk we will explore some of their algebraic and analytic properties.
Monday, July 12, 13:00 ~ 14:00 UTC-3
A CMV connection between orthogonal polynomials on the unit circle and the real line
María-José Cantero
Universidad de Zaragoza, España - This email address is being protected from spambots. You need JavaScript enabled to view it.
It is very well known the connection between orthogonal polynomials on the unit circle and orthogonal polynomials on the real line, given by Szeg\H{o}. 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 kinds 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 linear combination yields a Jacobi matrix ${\cal K}$ depending on a real parameter $\lambda$ ({\it general DVZ connection)}. The main result of the authors refers to the Jacobi matrix $\cal K$ built out of the Jacobi polynomials on the unit circle, which leads to a connection with the so called big $-1$ Jacobi polynomials. They also obtain the relation between the orthogonal polynomials and orthogonality measures for an arbitrary CMV matrix $\cal C$ and the corresponding Jacobi matrix $\cal K$, but only for the value $\lambda=1$, which simplifies the connection ({\it basic DVZ connection}). However, for generalized DVZ, such a general relation 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 (Universidad Carlos III de Madrid, España), Leandro Moral (Universidad de Zaragoza, España) and Luis Velázquez (Universidad de Zaragoza, España).
Monday, July 12, 14:00 ~ 15:00 UTC-3
$q$-Fractional Askey--Wilson Integrals and Related Semigroups of Operators
Mourad Ismail
University of Central Florida, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We introduce three one parameter semigroups of operators and determine their spectra. They are fractional integrals associated with the Askey--Wilson operator. We also study these families as families of positive linear approximation operators. Applications include connection relations and bilinear formulas for the Askey--Wilson polynomials. We also introduce a $q$-Gauss--Weierstrass transform and prove a representation and inversion theorem for it.
Joint work with Ruiming Zhang and Keru Zhou..
Monday, July 12, 15:00 ~ 16:00 UTC-3
Signal processing miracles and the Korteweg-de Vries equation.
F. Alberto Grunbaum
UC Berkeley, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We show that the master symmetries of the KdV equation give a way to extend the remarkable result of David Slepian (1964) in connection with the Bessel integral kernel and the existence of a differential operator that commutes with it. The original result of Slepian plays an important role in signal processing as well as in Random Matrix theory.
Tuesday, July 13, 16:00 ~ 17:00 UTC-3
An algebraic treatment of the Askey biorthogonal polynomials on the unit circle
Luc Vinet
CRM, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
A joint algebraic interpretation of the biorthogonal Askey polynomials on the unit circle and of the orthogonal Jacobi polynomials will be presented. That this ties their bispectral properties to an algebra called the meta-Jacobi algebra $m\mathfrak{J}$ will be explained.
Joint work with Alexei Zhedanov (Renmin University, China).
Tuesday, July 13, 17:00 ~ 18:00 UTC-3
Unified construction of all hypergeometric and basic hypergeometric orthogonal polynomial sequences.
Luis Verde-Star
Universidad Autónoma Metropolitana, Iztapalapa, México - This email address is being protected from spambots. You need JavaScript enabled to view it.
We present a construction of a class $H$ of polynomial sequences that satisfy a three-term recurrence relation and are eingenfunctions of a generalized difference equation of order one with respect to a Newton basis. The class $H$ contains all the hypergeometric and basic hypergeometric orthogonal polynomial sequences, and the sequences obtained with $q=-1$.
All the polynomial sequences in $H$ are determined by three linearly recurrent sequences of numbers that satisfy a difference equation of order three. Using the initial values of such sequences as parameters we obtain a uniform parametrization of all the families in the class. The parameters also provide alternative descriptions of the Askey and the $q$-Askey schemes.
Tuesday, July 13, 18:00 ~ 19:00 UTC-3
Relation between a class of Sobolev orthogonal polynomials on the unit circle and a subclass of the continuous dual Hahn polynomials
Cleonice Bracciali
UNESP - Univ Estadual Paulista, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work we deal with an example of Sobolev orthogonal polynomials on the unit circle and the associated connection coefficients. Under certain conditions we show that there is a relation between the associated connection coefficients and a subclass of the continuous dual Hahn polynomials.
Joint work with Jéssica V. da Silva (UNESP - Univ Estadual Paulista, Brazil) and A. Sri Ranga (UNESP - Univ Estadual Paulista, Brazil).
Tuesday, July 13, 19:00 ~ 20:00 UTC-3
Some results related to Bispectral Functions.
Ignacio Zurrian
CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we will discuss the role of bispectral functions in building local operators that commute with global operators (such as differential operators and integral operators). From the original motivations (e.g. time- and band-limiting) to more recent notions like reflecting operators, we will survey the importance of bispectrality and/or bispectral functions including many different set-ups.
Tuesday, July 13, 20:00 ~ 21:00 UTC-3
Stochastic factorizations of birth-death chains and Darboux transformations
Manuel D de la Iglesia
Universidad Nacional Autónoma de México, México - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $P$ be the transition probability matrix of a discrete-time birth-death chain, i.e. a stochastic Jacobi matrix. We consider factorizations of the form $P=P_1P_2$, where $P_1$ and $P_2$ are also stochastic matrices. By inverting the order of multiplication (also known as a Darboux transformation) we obtain new discrete-time birth-death chains $\widetilde{P}=P_2P_1$ from which we can identify the spectral measures associated with $\widetilde{P}$ from the original spectral measure and the corresponding orthogonal polynomials. We show several situations for different state spaces.
Joint work with F. A. Grünbaum (University of California, Berkeley) and C. Juarez (Universidad Nacional Autónoma de México).