Session III.7 - Special Functions and Orthogonal Polynomials
Talks
Monday, June 19, 14:00 ~ 14:30
Stieltjes–Fekete problem, degenerate orthogonal polynomials and Painlevé equations
Tamara Grava
SISSA/University of Bristol, Italy/UK - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study the set of double eigenvalues of a one parameter family of quartic/sextic anharmonic oscillators and we show that, under suitable rescaling, they behave as the poles of rational solutions of Painleve' equations.
Joint work with 1. Marco Bertola and Eduardo Chavez Heredia and 2. Marco Bertola and Dmitry Rachenkov.
Monday, June 19, 14:30 ~ 15:00
Zeros of Meixner polynomials for nonstandard parameter values
Kerstin Jordaan
University of South Africa, South Africa - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk I will prove a conjecture in [1] on a lower bound for the first positive zero of a Meixner polynomial when $-\beta,c\in(0,1)$ and use this to identify upper and lower bounds for the first few zeros of Meixner polynomials when $-2 \lt \beta \lt -1$ and $0 \lt c \lt 1$. I will also discuss other properties of the zeros of Meixner polynomials for these parameter ranges.
[1] K. Driver and A. Jooste, Quasi-orthogonal Meixner polynomials, Quaest. Math. 40 (4) (2017), 477-490
Joint work with Alta Jooste (University of Pretoria).
Monday, June 19, 15:00 ~ 15:30
Asymptotics of Fredholm determinants associated with higher dimensional kernels
Dan Dai
City University of Hong Kong, Hong Kong - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we consider the Pearcey kernel and the hard edge Pearcey kernel, which appear in random matrix theory and many other stochastic models. They are viewed as higher dimensional kernels in the sense that they can be characterized by 3 × 3 matrix-valued Riemann-Hilbert problems. We establish integral representations for Fredholm determinants of integral operators with these kernels, which involve Hamiltonians associated with certain nonlinear differential equations. We also derive large gap asymptotics for the determinants and obtain asymptotic statistical properties for the related point processes.
Joint work with Shuai-Xia Xu (Sun Yat-sen University, China) and Lun Zhang (Fudan University, China).
Monday, June 19, 15:30 ~ 16:00
On strong asymptotics of multiple orthogonal polynomials for Angelesco systems
Maxim Yattselev
Indiana University - Purdue University Indianapolis, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will discuss recent results on strong asymptotics of multiple orthogonal polynomials for Angelesco systems.
Monday, June 19, 16:30 ~ 17:30
Orthogonal Polynomials and Symmetric Freud weights
Peter Clarkson
University of Kent, UK - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk I will discuss orthogonal polynomials associated with symmetric Freud weights, in particular the sextic weight \[ \omega(x;t,\tau,\rho)=|x|^\rho\exp(-x^6+\tau x^4+tx^2),\eqno(1)\] with $\tau$, $t$ and $\rho \gt -1$ parameters. I will describe properties of the recurrence coefficients in the three-term recurrence relation associated with these orthogonal polynomials. For the sextic weight (1) the recurrence coefficients satisfy a fourth-order discrete equation which is the second member of the first discrete Painlevé hierarchy, also known as the string equation, and also satisfiy a coupled system of second-order, nonlinear differential equations. When $\rho=0$, the weight (1) arises in the context of Hermitian matrix models and random symmetric matrix ensembles.
Joint work with Kerstin Jordaan (University of South Africa, South Africa) and Ana Loureiro (University of Kent, UK).
Monday, June 19, 17:30 ~ 18:00
Hidden Algebraic Structures in Expansions Relating to Airy, Bessel and Painlevé Functions
Folkmar Bornemann
Technische Universität München, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Recent work on the asymptotic expansion of the hard-to-soft transition limit in random matrix theory has revealed unexpected algebraic structures relating to Airy, Bessel and Painlevé Functions: e.g., divisibility properties of certain polynomials relating to Olver's asymptotic expansion of Bessel functions of large order in the transition region, or expansion terms of operator determinants (tau functions of Painlevé equations) being linear combinations of higher order derivatives of the Tracy-Widom distributions with rational polynomial coefficients. So far, all these structures were found only algorithmically based on CAS software, or in parts only numerically, by inspecting the first few (say, 10 to 100) accessible concrete cases. Proofs will be challenging, since these structures must be related to some underlying "integrability".
Monday, June 19, 18:00 ~ 18:30
Exponential asymptotics for Airy solutions of Painlevé II
Alfredo Deaño
Universidad Carlos III de Madrid, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we present recent results on exponential asymptotics for Airy solutions of the Painlevé II equation, as the variable tends to infinity. The starting point is the leading term in the asymptotic expansion of the corresponding tau function, which can be obtained using a multiple integral formulation and the classical method of steepest descent. Using an exponential asymptotics ansatz for the tau function (which is a one parameter case of known general transseries for solutions of Painlevé II), we can put this family of solutions in the framework of exponential asymptotics , work out higher order terms and also use these for precise evaluation of the zeros of the tau function in the complex plane.
Joint work with Inês Aniceto (University of Southampton, United Kingdom) and Roberto Vega Álvarez (Instituto Superior Técnico, Lisbon, Portugal).
Tuesday, June 20, 14:00 ~ 14:30
Computation of confluent hypergeometric functions
Amparo Gil
Universidad de Cantabria, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
Confluent hypergeometric functions appear in many applications in applied mathematics, physics and engineering. Despite their importance, few algorithms are available for computing any of the standard solutions of Kummer's equation in the case of real or complex parameters. In this talk, we present recent advances in the computation of the Kummer function $U(a,b,x)$ for real values of the parameters [1] and in the evaluation of a particular case of the Kummer function, the parabolic cylinder function $U(a,z)$ for complex arguments [2].
On the other hand, confluent hypergeometric functions play a key role in the asymptotic analysis of Fermi-Dirac integrals. The evaluation of these integrals and their derivatives is crucial for computations in physical problems dominated by the degenerate fermions. Therefore, the Fermi-Dirac integrals are widely used in stellar astrophysics or plasma physics. In this lecture we will show that the use of asymptotic expansions allow to calculate these integrals efficiently and with high accuracy for a large range of parameters.
References
[1] A. Gil, D. Ruiz-Antolín, J. Segura, N.M. Temme. Efficient and accurate computation of confluent hypergeometric function $U(a,b,x)$. Submitted.
[2] T.M. Dunster, A. Gil, J. Segura. Computation of parabolic cylinder functions having complex argument. Submitted.
[3] A. Gil, J. Segura, N.M. Temme. Complete asymptotic expansions for the relativistic Fermi-Dirac integral. Appl. Math. Comput. 412 (2022) 126618.
[4] A. Gil, A. Odrzywolek, J. Segura, N.M. Temme. Evaluation of the relativistic Fermi-Dirac integral and its derivatives for moderate/large values of the parameters. Comput. Phys. Commun. 283 (2023) 108563.
Joint work with T. M. Dunster (San Diego State University, USA), J. Segura (Universidad de Cantabria, Spain) and N.M. Temme (CWI, The Netherlands).
Tuesday, June 20, 14:30 ~ 15:00
On a new class of classical 2-orthogonal polynomials
Khalfa Douak
Sorbonne Université - Campus Pierre et Marie Curie, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, I will present recent results on 2-orthogonal polynomials obtained in two joint papers with Pascal Maroni [1], [2]. This constitutes a few contribution in the area of the $d$-orthogonal polynomials. Thanks to the Hahn property defining the classical character, we first give a new class of classical 2-orthogonal polynomials for which we determine the explicit expressions of the recurrence coefficients with some other properties. Six special cases of those polynomials have been pointed out. The aim of the second part is to seek integral representations for the pair of linear functionals with respect to which such polynomials are 2-orthogonal. We start with the matrix differential equation satisfied by the vector of those functionals which, in turn, provides differential equations satisfied by the respective (sought) weight functions. Depending on the case, we obtain that these weights are defined in terms of various special functions with supports on the real line or on positive real line. In order for certain integral representations to exist, addition of Dirac masses at the origin is necessary.
References:
[1] K. Douak , P. Maroni. On a new class of 2-orthogonal polynomials, I: The recurrence relations and some properties. Integral Trans. Spec. Funct., 32(2) (2021) 134--153.
[2] K. Douak , P. Maroni. On a new class of 2-orthogonal polynomials, II: The integral representations, arXiv:2212.11949v2 (2023) 1--30.
Joint work with Pascal Maroni.
Tuesday, June 20, 15:00 ~ 16:00
Exceptional orthogonal polynomials and isospectral deformations
Robert Milson
Dalhousie University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
Exceptional ortogonal polynomials are solutions of second-order Sturm-Liouville problems. However, unlike their classical counter-parts the families in questions consist of polynomials that are missing a finite number of degrees. I will report on some recent work that allows for the construction of excpetional Jacobi polynomials with an aribtrary number of continuous parameters. These parameters serve as deformation parameters for isospectrally equivalent families of self-adjoint operators. Time permitting, we will describe the role of these new construction in the ongoing classification project for exceptional orthogonal polynomials.
Joint work with David Gomez-Ullate, MariaAngeles Garcia Ferrero.
Tuesday, June 20, 16:30 ~ 17:00
Exceptional polynomials and how to find them
María Ángeles García-Ferrero
Universitat de Barcelona, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
Exceptional orthogonal polynomials arise as eigenfunctions of Sturm-Liouville problems and form complete bases in Hilbert spaces despite the fact of missing some degrees. Since 2009, many pages have been written about their construction and their properties, but the book is still unfinished.
In this talk, we will review some of the last lines added to the narrative of exceptional polynomials and what is missed to finish the corresponding chapters.
Joint work with D. Gómez-Ullate (IE University, Spain) and R. Milson (Dalhousie University, Canada).
Tuesday, June 20, 17:00 ~ 17:30
Lattice paths, branched continued fractions, and multiple orthogonal polynomials
Helder Lima
KU Leuven, Belgium - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will present an overview of the recently found connection between multiple orthogonal polynomials and branched continued fractions that arise as generating functions of lattice paths and were introduced to solve total-positivity problems of combinatorial interest. Production matrices and total positivity play an important role in this connection. I start by giving a brief introduction to the different topics involved in this talk and then I explain how the study of their connection brings to light new results on different fields, with emphasis on the results about multiple orthogonal polynomials.
Tuesday, June 20, 17:30 ~ 18:00
An approach to universality using Weyl m-functions
Brian Simanek
Baylor University, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will present new results on scaled limits of polynomial reproducing kernels for probability measures on the real line and unit circle. These limits are known as universality limits and we will focus on the case when one obtains the sine kernel in the limit. Our results generalize much of the previous literature and show that universality at a point is a local property of the measure. Our approach uses a matrix version of the Christoffel-Darboux kernel.
Joint work with Benjamin Eichinger (Vienna University of Technology) and Milivoje Lukic (Rice University).
Tuesday, June 20, 18:00 ~ 18:30
Special functions and unitarity
Erik Koelink
Radboud University, the Netherlands - This email address is being protected from spambots. You need JavaScript enabled to view it.
Orthogonal polynomials and special functions often behave well with respect to certain integral transforms which can be viewed as unitary operators on a Hilbert space of functions. A classic example is the Fourier transform mapping Hermite functions to Hermite functions. Such integral transforms can also be used to map `simpler' special functions to more `complicated' ones. There are several approaches to understand these transforms, such as e.g. via the study of the kernels or via the study of an appropriate self-adjoint operator on a suitable Hilbert space of functions. We discuss these approaches and give several examples.
Wednesday, June 21, 14:00 ~ 14:30
Matrix valued orthogonal polynomials and Darboux factorizations
Pablo Roman
Universidad Nacional de Cordoba, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
Matrix valued orthogonal polynomials (MVOPs) which are eigenfunctions of a second order differential operator have been studied recently. This has led to the extension of numerous results of the classical scalar theory. In particular, the so called matrix Bochner problem, solved by Casper and Yakimov, gives a classification of the families with this property. However, the explicit construction of families of arbitrary size is still a difficult problem.
In this talk, we will discuss Darboux factorizations of second order differential operators associated to a family of MVOPs. We will show that such a factorization can be used to construct new families. In the simplest case, this construction leads to lowering and rising shift operators for matrix valued orthogonal polynomials. More general factorizations lead to matrix valued analogues of exceptional polynomials. We will discuss the main properties and explicit examples.
Joint work with Joint work with E. Koelink and Lucia Morey..
Wednesday, June 21, 14:30 ~ 15:30
Generalized polynomials and generalized Gaussian quadrature
Daan Huybrechs
KU Leuven, Belgium - This email address is being protected from spambots. You need JavaScript enabled to view it.
Orthogonal polynomials are linearly independent functions, which form a so-called Chebyshev set. Chebyshev sets are defined by uniqueness of interpolation: like polynomials, a Chebyshev set with n functions has a unique solution to the interpolation problem in any set of n distinct points. For this reason, expansions in Chebyshev sets are sometimes called generalized polynomials. It turns out that the unique interpolation property is sufficient to define Gaussian quadrature, in this context called generalized Gaussian quadrature, for the numerical evaluation of integrals. The rules are Gaussian in the sense that with just n points they are exact for twice as many basis functions. Surprisingly, one does not need orthogonality or other algebraic properties of polynomials for this result to be true. For classical Gaussian quadrature, the points are the roots of an orthogonal polynomial. What are they in the case of generalized polynomials? We describe a theoretical framework and formulate a practical and convergent algorithm to compute the rules. We illustrate the method with novel numerical examples of generalized Gaussian quadrature and applications.
Wednesday, June 21, 15:30 ~ 16:00
Sobolev orthogonal polynomials for solving the Schrödinger equation with potentials $V (x) = x^{2k}$, $k \geqslant 2$.
Teresa E. Pérez
Universidad de Granada, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
The variational formulation of a boundary value problem for the harmonic oscillator with potentials $V (x) = x^{2k}$, $k \geqslant 2$, given by \begin{align*} -&u'' + \lambda \,x^{2k}\,u = f(x),\\ &u(-1) = u(1) =0, \end{align*} for $\lambda \gt 0$, provides a Sobolev inner product, that is, an inner product involving derivatives that can be deeply studied, and used to solve the BVP.
In this work we study the associated family of Sobolev orthogonal polynomials, including a recursive way to compute them as well as the outer relative asymptotics of this polynomials and classical Legendre polynomials.
The analysis of the Fourier-Sobolev coefficients and some numerical experiments complete this talk.
Joint work with Lidia Fernández (Universidad de Granada, Spain), Francisco Marcellán (Universidad Carlos III, Spain) and Miguel A. Piñar (Universidad de Granada, Spain).
Wednesday, June 21, 16:30 ~ 17:00
On some properties of a family of Jacobi Polynomials arising in the evaluation of integrals
John Lopez
Tulane University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk I will discuss some properties related to the Jacobi Polynomials with non classical parameters of the form $\alpha_m=m+1/2$ and $\beta_m=-m-1/2$. First I will present two different approaches for the computation of the moments, one of them is an explicit computation by parametrization of the contour, and the other is by using residue Theorem. I will present a relation between the moments and the generating function of the weights. Finally, I will discuss the asymptotic behavior of the zeros of these polynomials, which is related to a limit case of the results found in [KM04].
These problems appeared in the search for the definite integral of the negative power of a quartic polynomial.
\textbf{References}.
[BM 99] G. Boros, V.H. Moll, An integral hidden in Gradshteyn and Ryzhik, J., \textit{Appl. Math}. \textbf{106} (1999) 361–368.
[DM01] K. Driver, M. M\"olelr, Zeros of hypergeometric polynomials $F(-n,b,-2n;z)$, \textit{J. Approx. Theory}, \textbf{110} (2001), 74-87.
[KM04] A.B.J. Kuijlaars, A Martínez-Finkelshtein, Strong asymptotics for Jacobi polynomials with varying nonstandard parameters, \textit{Journal d’analyse Math{\'e}matique}. Springer \textbf{94} (2004),195-234.
[KMO05] A.B.J. Kuijlaars, A. Martínez-Finkelshtein, R. Orive, Orthogonality of Jacobi polynomials with general parameters, \textit{Elect. Trans. in Numer. Anal} \textbf{19} (2005) 1-17
[SKF15] Patrick Njionou Sadjang, Wolfram A. Koepf and Mama Foupouagnigni, On Moments of Classical Orthogonal Polynomials, \textit{J. Mat Anal. Appl}. 424. (2015) 122-151. no 1, 122151.
Joint work with Victor Moll and Kenneth McLaughlin.
Wednesday, June 21, 17:00 ~ 17:30
About 2-orthogonal polynomial eigenfunctions of a third order differential operator
Teresa A. Mesquita
Instituto Politécnico de Viana do Castelo & Centro de Matemática da Universidade do Porto, Portugal - This email address is being protected from spambots. You need JavaScript enabled to view it.
The $d$-orthogonal polynomial sequences are known to fulfil certain differential equations of order $d+1$ (e.g. [1, 2, 3]). Considering a generic third order differential operator that does not increase the degree of polynomials, as expressed in [4], we present explicit descriptions of corresponding 2-orthogonal polynomial eigenfunctions. Furthermore, their Hahn-classical character is analysed and other differential identities are given as a consequence of the symbolic approach used in this research work.
REFERENCES
[1] K. Douak; The relation of the d-orthogonal polynomials to the Appell polynomials; J. Comput. Appl. Math. 70(2), 279-295 (1996).
[2] K. Douak and P. Maroni; On d-orthogonal Tchebyshev polynomials, I ; Appl. Num. Math., 24, 23-53 (1997).
[3] H. Lima and A. Loureiro; Multiple orthogonal polynomials associated with confluent hypergeometric functions; J. Approx. Theory 260, 36 p. (2020).
[4] T. A. Mesquita and P. Maroni; Around operators not increasing the degree of polynomials; Integral Transforms Spec. Funct. 30, No.5, 383-399 (2019).
[5] T. A. Mesquita; Symbolic Approach to 2-Orthogonal Polynomial Solutions of a Third Order Differential Equation; Math.Comput.Sci. (DOI : 10.1007/s11786-022-00525-8)
Wednesday, June 21, 17:30 ~ 18:00
A least squares analog to the Nuttall-Pommerenke theorem.
Laurent Baratchart
INRIA centre de l'Université de Nice, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
In various contexts involving identification and design, the following least-squares substitute to multipoint Padé approximation became quite popular in recent years under the name of "vector fitting": given a holomorphic function $f$ and a set of points $z_1,\cdots,z_N$ in the complex plane, to find a rational function $p_m/q_n$ of type $(m,n)$ minimizing the criterion $\sum_{j=1}^N |q(z_j)f(z_j)-p(z_j)|^2$. This type of approximation involves non-classical orthogonality, and its behaviour is still fairly open.
We analyze here the classical Padé analog where one minimizes the $l^2$-norm of the first n+m+1 terms of the Taylor expansion of the linearized error at a point $z_0$. In particular, we prove that convergence in capacity prevails when $f$ is analytic on the complex plane minus a polar set; i.e., a set of logarithmic capacity zero, provided that $N\leq C(n+m)$. This least-square version of the Nutall-Pommerenke theorem also sheds light on the multipoint case.
Joint work with Paul Asensio.
Wednesday, June 21, 18:00 ~ 18:30
Applications of multiple orthogonal polynomials associated with moments that are ratios of products of gamma functions.
Thomas Wolfs
KU Leuven, Belgium - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will discuss two applications of multiple orthogonal polynomials associated with moments that are ratios of products of gamma functions. The first one is in random matrix theory (squared singular values of products of random matrices) and the other one in Diophantine approximation problems. Recently, such polynomials also appeared in connection to the theory of branched continued fractions. The properties of particular cases of these polynomials have been studied throughout the years, e.g. certain Jacobi-Piñeiro polynomials by [Smet-Van Assche, 2010], the multiple orthogonal polynomials associated with the K-Bessel functions by [Van Assche-Yakubovich, 2000], with the confluent/Gauss' hypergeometric functions by [Lima-Loureiro, 2020/2021] and with the exponential integral by [Van Assche-Wolfs, 2022].
Joint work with Walter Van Assche (KU Leuven, Belgium).
Posters
Rate of Pole Detection Using Pad\'{e} Approximants to Polynomial Expansions
Nattapong Bosuwan
Mahidol University , Thailand - This email address is being protected from spambots. You need JavaScript enabled to view it.
Pad\'{e} approximation and its generalizations were proved to be useful tools in localization of singularities of functions. In this work, we investigate the rate of detection of poles using orthogonal Pad\'{e} and Pad\'{e}-Faber approximations. We provide global estimates on this rate of detection and show that our global estimates are sharp. Our results in this paper serve as an analogue of the direct part of Gonchar's theorem in 1981 for such Pad\'{e} approximants.
Joint work with Methawee Wajasat (Mahidol University).
Symmetry structure of starbursts
Antonia M. Delgado
Universidad de Granada, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
When looking at a star under low-light conditions, most people perceive some structured symmetric patterns, which have been called starbursts. Starburst patterns can be very diverse but some typical ones are those in which a bright central area is surrounded by clearly marked intensity spikes (star points) [2, 3]. These light patterns are formed due to the imperfections in the optical elements of the human eye, which are mathematically described by a bivariate orthogonal Zernike expansion. Based on the deep relation between wavefront aberration and caustic patterns symmetry-preserving and the properties of some singular points of curvature functions of the classical Zernike polynomials [1], in this work we investigate a theoretical explanation of the types of symmetries and the number of points of starbursts.
References
[1] S. Barbero, A. Bradley, N. López-Gil, J. Rubinstein, L. Thibos, Catastrophe optics theory unveils the localised wave aberration features that generate ghost images, Ophthalmic Physiol Opt. 2022; 42: 1074-1091.
[2] J. Rubinstein, On the geometry of visual starbursts, J. Opt. Soc. Am. A 36(4), B58-B64 (2019).
[3] R. Xu, L. N. Thibos, N. López-Gil, P. Kollbaum, A. Bradley, Psychophysical study of the optical origin of starbursts, J. Opt. Soc. Am. A 36(4), B97-B102 (2019).
Joint work with Sergio Barbero (CSIC - Instituto de Óptica Daza de Valdés, España), Lidia Fernández (Universidad de Granada, España) and Teresa E. Pérez (Universidad de Granada, España).
Hahn multiple orthogonal polynomials of type I
Juan Díaz
Universidade de Aveiro, Portugal - This email address is being protected from spambots. You need JavaScript enabled to view it.
Explicit expressions for the Hahn multiple polynomials of type I, in terms of Kampé de Fériet hypergeometric series, are given. Orthogonal and biorthogonal relations are proven. Then, part of the Askey scheme for multiple orthogonal polynomials type I is completed. In particular, explicit expressions in terms of generalized hypergeometric series and Kampé de Fériet hypergeometric series, are given for the multiple orthogonal polynomials of type I for the Jacobi-Piñeiro, Meixner I, Meixner II, Kravchuk, Laguerre I, Laguerre II and Charlier families.
Joint work with Amílcar Branquinho (Universidade de Coimbra, Portugal), Ana Foulquié (Universidade de Aveiro, Portugal) and Manuel Mañas (Universidad Complutense de Madrid, Spain).
Laguerre matrix orthogonal polynomials: A Riemann-Hilbert Approach
Assil Fradi
Universidade de Aveiro, Portugal - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider matrix orthogonal polynomials related to Laguerre type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann–Hilbert problem we can derive first and second order differential relations satisfied by matrix orthogonal polynomials and the associated second kind functions . Finally, non-Abelian extensions of a family of discrete Painlevé d-PIV equations are obtained for the three term recurrence relation coefficients.
Joint work with AMÍLCAR BRANQUINHO (Universidade de Coimbra, Portugal), ANA FOULQUIÉ-MORENO (Universidade de Aveiro, Portugal) and MANUEL MAÑAS (Universidad Complutense de Madrid, Spain).
Moments of Jacobi Polynomials with non-classical parameters
John Lopez
Tulane, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this poster we will present two different approaches to the computation of the moments of Jacobi Polynomials with non classical parameters of the form $\alpha_m=m+1/2$, $\beta_m=-m-1/2$. Both of them start from defining the weight function $w(z, \alpha_m, \beta_n)=(1-x)^{\alpha_m}(1+x)^{\beta_m}$ continuously in a contour $\Gamma$ on a Riemann Surface. The first approach is an explicit computation by parametrizing $\Gamma$, and the second is by using residues on a closed curve obtained by deforming the contour $\Gamma$ and performing a change of variables. This problems appeared in the search for the definite integral of the negative power of a quartic polynomial.
References:
G. Boros, V.H. Moll, An integral hidden in Gradshteyn and Ryzhik, J., Appl. Math. 106 (1999) 361–368.
A.B.J. Kuijlaars, A. Martínez-Finkelshtein, R. Orive, Orthogonality of Jacobi polynomials with general parameters, Elect. Trans. in Numer. Anal 19 (2005) 1-17.
Patrick Njionou Sadjang, Wolfram A. Koepf and Mama Foupouagnigni, On Moments of Classical Orthogonal Polynomials, J. Mat Anal. Appl. 424. (2015) 122-151. no 1, 122151.
Joint work with Victor Moll (Tulane University, US) and Kenneth McLaughlin (Tulane University, US).
Mehler-Heine asymptotics for $q$-hypergeometric polynomials
Juan F. Mañas-Mañas
Universidad de Almería, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
The basic $q$-hypergeometric function $ _r\phi_s$ is defined by the series
$$ _r\phi_{s}\left(\begin{array}{l} a_{1}, \ldots, a_{r} \\ b_{1}, \ldots, b_{s} \end{array} ; q, z\right) =\sum_{k=0}^{\infty} \frac{\left(a_{1}; q\right)_{k}\cdots\left(a_{r} ; q\right)_{k}}{\left(b_{1}; q\right)_{k}\cdots\left(b_{s} ; q\right)_{k}}\left((-1)^kq^{\binom{k}{2}}\right)^{1+s-r}\frac{z^{k}}{(q ; q)_{k}}, $$ where $0 \lt q \lt 1$ 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 $ 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.
Concretely, by scaling adequately these polynomials we 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 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 deduce a similar relation in the context of the $q$-analysis and we will illustrate this with numerical examples. The results have recently been published in [1].
\noindent [1] J.F. Ma\~{n}as--Ma\~{n}as, J.J. Moreno--Balc\'{a}zar, Asymptotics for some $q$-hypergeometric polynomials, Results Math. 77(4), Art. 146 (2022).
Joint work with Juan J. Moreno-Balcázar (Universidad de Almería, Spain).
Orthogonality of Polynomials Involved in a Linear Combination with Chebyshev Polynomials of the Second Kind
Mirela Vanina de Mello
Universidade Estadual de Santa Cruz - UESC, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $\{U_n\}_{n\geq0}$ and $\{S_n\}_{n\geq0}$ be sequences of polynomials such that $$U_n(x) = S_n(x) + a_{n-1}S_{n-1}(x),$$ $n \geq 1,$ where $\{a_n\}_{n\geq0} \in \mathbb{R}$ and $U_n$ are the orthogonal Chebyshev polynomials of the second kind. Our interest is to find out when $\{S_n\}_{n\geq0}$ is a sequence of orthogonal polynomials.
Marcellán and Petronilho [1] solved this problem by imposing conditions on the coefficients $a_n$. They also obtained a relationship between the linear functionals related to the orthogonal polynomials cited. Using results for recovery the orthogonality measure via Turán determinants [2], we determined both: the sequence of coefficients $a_n$ for which $\{S_n\}_{n\geq0}$ is orthogonal, and not only the linear functional, but also the weight function with respect to which the corresponding polynomials $S_n$ are orthogonal. In other words, the answer to the question posed above was obtained in a completely different and independent way from the approach of Marcellán and Petronilho, with our approach being analytical while the other is entirely algebraic.
Bibliography
[1] F. Marcellán; J. Petronilho, Orthogonal polynomials and coherent pairs: the classical case, Indag. Mathem. 6 (1995), 287-307.
[2] A. Máté; P. Nevai; V. Totik, Asymptotics for orthogonal polynomials defined by a recurrence relation, Constr. Approx. 1 (1985) 231-248.
Symbolic computation of Mehler-Heine asymptotics for Sobolev-type orthogonal polynomials
Juan J. Moreno-Balcázar
Universidad de Almería, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this poster we tackle the Mehler-Heine formulae for Sobolev-type orthogonal polynomials. We present the theoretical results and an algorithm to compute these formulae effectively. The algorithm allows us to construct a computer program based on Mathematica® language, where the corresponding Mehler-Heine formulae are automatically obtained. Applications and examples show the efficiency of the approach developed.
These results have been published in [1] and the Mathematica® code is available for free in [2].
[1] Sobolev orthogonal polynomials: asymptotics and symbolic computation, J. F. Mañas-Mañas, J. J. Moreno-Balcázar, East Asian J. Appl. Math. 12(3), 535-563, 2022.
[2] Notebook Archive (2022), https://notebookarchive.org/2022-06-amlp3fh
Joint work with Juan F. Mañas-Mañas (Universidad de Almería, Spain).
Approximate Calculation of Sums based on a Gaussian type quadrature
Vanessa Paschoa
Universidade Federal de São Paulo, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.
The best approximation polynomial with respect to a table with large amount of data by the method of least squares can be given, in a stable way, in an orthogonal basis. To find this polynomial is necessary to calculate numerical sums with large amount of terms. We proposed an algorithmic approach for approximate calculation of sums of the form $\sum_{j=1}^N f(j)$ based on a Gaussian type quadrature formula for sums. This allows the calculation of sums with very large number of terms $N$ to be reduced to sums with a much smaller number of summands $n$. For use the Gaussian type quadrature we need the $n$ nodes that are the zeros of Gram polynomials, also known as Discrete Chebyshev polynomials. In [1] we obtained precise lower and upper bounds for these zeros. We proved in [2] that the Weierstrass--Dochev--Durand--Kerner iterative numerical method, with explicitly given initial conditions, converges to the nodes of the quadrature formula. Several methods for computing the nodes of the discrete analogue of the Gaussian quadrature formula are compared. Since, for practical purposes, any approximation of a sum should use only the values of the summands, we implement a simple but efficient procedure to additionally approximate the evaluations at the nodes by local natural splines. Explicit numerical examples are provided. Moreover, the error in different spaces of functions was analyzed.
Bibliography [1] Area, I., Dimitrov, D. K., Godoy E., Paschoa, V. G. . Approximate Calculation of Sums I: Bounds for the Zeros of Gram Polynomials. {\it SIAM Journal on Numerical Analysis}, v. 52, p. 1867-1886, 2014.
[2] Area, I., Dimitrov, D. K., Godoy E., Paschoa, V. G. Approximate Calculation of Sums II: Gaussian Type Quadrature. {\it SIAM Journal on Numerical Analysis}, v. 54, p. 2210-2227, 2016.
Joint work with Iván Area (Universidad de Vigo, Spain), Dimitar K. Dimitrov (Universidade Estadual Paulista Júlio de Mesquita Filho, Brazil) and Eduardo Godoy (Universidad de Vigo, Spain)..
Non-linear difference equations for complex Verblunsky coefficients of orthogonal polynomials on the unit circle
Karina Rampazzi
Universidade Estadual Paulista (UNESP), Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider orthogonal polynomials on the unit circle associated with certain semiclassical weight functions.The Pearson-type differential equations satisfied by these weight functions involve two polynomials of degree at most 2. General structure relations for the orthogonal polynomials and non-linear difference equations for the associated complex Verblunsky coefficients are presented. As an application, we present several new structure relations and non-linear difference equations associated with a special weight function, which is an extension of the circular Jacobi weight function.
Joint work with Cleonice F. Bracciali (Universidade Estadual Paulista - Brazil) and Luana L. Silva Ribeiro (Universidade de Brasilia - Brazil).