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2020 | Buch

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Orthogonal Polynomials

2nd AIMS-Volkswagen Stiftung Workshop, Douala, Cameroon, 5-12 October, 2018

herausgegeben von: Prof. Dr. Mama Foupouagnigni, Wolfram Koepf

Verlag: Springer International Publishing

Buchreihe : Tutorials, Schools, and Workshops in the Mathematical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book presents contributions of international and local experts from the African Institute for Mathematical Sciences (AIMS-Cameroon) and also from other local universities in the domain of orthogonal polynomials and applications. The topics addressed range from univariate to multivariate orthogonal polynomials, from multiple orthogonal polynomials and random matrices to orthogonal polynomials and Painlevé equations.

The contributions are based on lectures given at the AIMS-Volkswagen Stiftung Workshop on Introduction of Orthogonal Polynomials and Applications held on October 5–12, 2018 in Douala, Cameroon. This workshop, funded within the framework of the Volkswagen Foundation Initiative "Symposia and Summer Schools", was aimed globally at promoting capacity building in terms of research and training in orthogonal polynomials and applications, discussions and development of new ideas as well as development and enhancement of networking including south-south cooperation.

Inhaltsverzeichnis

Frontmatter

Introduction to Orthogonal Polynomials

Frontmatter

An Introduction to Orthogonal Polynomials

Abstract

In this introductory talk, we first revisit with proof for illustration purposes some basic properties of a specific system of orthogonal polynomials, namely the Chebyshev polynomials of the first kind. Then we define the notion of orthogonal polynomials and provide with proof some basic properties such as: The uniqueness of a family of orthogonal polynomials with respect to a weight (up to a multiplicative factor), the matrix representation, the three-term recurrence relation, the Christoffel-Darboux formula and some of its consequences such as the separation of zeros and the Gauss quadrature rules.

Mama Foupouagnigni

Classical Continuous Orthogonal Polynomials

Abstract

Classical orthogonal polynomials (Hermite, Laguerre, Jacobi and Bessel) constitute the most important families of orthogonal polynomials. They appear in mathematical physics when Sturn-Liouville problems for hypergeometric differential equation are studied. These families of orthogonal polynomials have specific properties. Our main aim is to:

recall the definition of classical continuous orthogonal polynomials;

prove the orthogonality of the sequence of the derivatives;

prove that each element of the classical orthogonal polynomial sequence satisfies a second-order linear homogeneous differential equation;

give the Rodrigues formula.

Salifou Mboutngam

Generating Functions and Hypergeometric Representations of Classical Continuous Orthogonal Polynomials

Abstract

The aim of this work is to show how to obtain generating functions for classical orthogonal polynomials and derive their hypergeometric representations.

Maurice Kenfack Nangho

Properties and Applications of the Zeros of Classical Continuous Orthogonal Polynomials

Abstract

Suppose $\{P_n\}_{n=0}^\infty $ is a sequence of polynomials, orthogonal with respect to the weight function w(x) on the interval [a, b]. In this lecture we will show that the zeros of an orthogonal polynomial are simple, that they are located in the interval of orthogonality and that the zeros of polynomials with adjacent degree, separate each other. We will also discuss the main ingredients of the Gauss quadrature formula, where the zeros of orthogonal polynomials are of decisive importance in approximating integrals.

A. S. Jooste

Inversion, Multiplication and Connection Formulae of Classical Continuous Orthogonal Polynomials

Abstract

Our main objective is to establish the so-called connection formula,

$$\displaystyle \begin{aligned} p_n(x)=\sum_{k=0}^{n}C_{k}(n)y_k(x), \end{aligned} $$

(0.1)

which for p _n(x) = x ⁿ is known as the inversion formula

$$\displaystyle \begin{aligned} x^n=\sum _{k=0}^{n}I_{k}(n)y_k(x), \end{aligned} $$

for the family y _k(x), where $\{p_n(x)\}_{n\in \mathbb {N}_0}$ and $\{y_n(x)\}_{n\in \mathbb {N}_0}$ are two polynomial systems. If we substitute x by ax in the left hand side of (0.1) and y _k by p _k, we get the multiplication formula

$$\displaystyle \begin{aligned} p_n(ax)=\sum _{k=0}^{n}D_{k}(n,a)p_k(x). \end{aligned} $$

The coefficients C _k(n), I _k(n) and D _k(n, a) exist and are unique since deg p _n = n, deg y _k = k and the polynomials {p _k(x), k = 0, 1, …, n} or {y _k(x), k = 0, 1, …, n} are therefore linearly independent. In this session, we show how to use generating functions or the structure relations to compute the coefficients C _k(n), I _k(n) and D _k(n, a) for classical continuous orthogonal polynomials.

Daniel Duviol Tcheutia

Classical Orthogonal Polynomials of a Discrete and a q-Discrete Variable

Abstract

The classical orthogonal polynomials of discrete and q-discrete orthogonal polynomials are introduced from their difference and q-difference equations. Some structure formulas are proved for the Charlier and the Al-Salam Carlitz polynomials from their generating functions.

Patrick Njionou Sadjang

Computer Algebra, Power Series and Summation

Abstract

Computer algebra systems can do many computations that are relevant for orthogonal polynomials and their representations. In this preliminary training we will introduce some of those important algorithms: the automatic computation of differential equations and formal power series, hypergeometric representations, and the algorithms by Fasenmyer, Gosper, Zeilberger and Petkovšek/van Hoeij.

Wolfram Koepf

On the Solutions of Holonomic Third-Order Linear Irreducible Differential Equations in Terms of Hypergeometric Functions

Abstract

We present here an algorithm that combines change of variables, exp-product and gauge transformation to represent solutions of a given irreducible third-order linear differential operator L, with rational function coefficients and without Liouvillian solutions, in terms of functions $S\in \left \{{{ }_1F_1}^2, ~{{ }_0F_2}, ~_1F_2, ~_2F_2\right \}$ where _pF _q with p ∈{0, 1, 2}, q ∈{1, 2}, is the generalized hypergeometric function. That means we find rational functions r, r ₀, r ₁, r ₂, f such that the solution of L will be of the form

$$\displaystyle y=~ \exp \left (\int r \,dx \right )\left (r_0S(f(x))+r_1(S(f(x)))^{\prime }+r_2(S(f(x)))^{\prime \prime }\right ). $$

An implementation of this algorithm in Maple is available.

Merlin Mouafo Wouodjié

The Gamma Function

Abstract

After the so-called elementary functions as the exponential and the trigonometric functions and their inverses, the Gamma function is the most important special function of classical analysis. In this note, we present the definition and properties of the Gamma and the Beta functions.

Daniel Duviol Tcheutia

Recent Research Topics in Orthogonal Polynomials and Applications

Frontmatter

Hypergeometric Multivariate Orthogonal Polynomials

Abstract

In this lecture a comparison between univariate and multivariate orthogonal polynomials is presented. The first step is to review classical univariate orthogonal polynomials, including classical continuous, classical discrete, their q-analogues and also classical orthogonal polynomials on nonuniform lattices. In all these cases, the orthogonal polynomials are solution of a second-order differential, difference, q-difference, or divided-difference equation of hypergeometric type. Next, a review multivariate orthogonal polynomials is presented. In the approach we have considered, the main tool is the partial differential, difference, q-difference or divided-difference equation of hypergeometric type the polynomial sequences satisfy. From these equations satisfied, the equation satisfied by any derivative (difference, q-difference or divided-difference) of the polynomials is obtained. A big difference appears for nonuniform lattices, where bivariate Racah and for bivariate q-Racah polynomials satisfy a fourth-order divided-difference equation of hypergeometric type. From this analysis, we propose a definition of multivariate classical orthogonal polynomials. Finally, some open problems are stated.

Iván Area

Signal Processing, Orthogonal Polynomials, and Heun Equations

Abstract

A survey of recents advances in the theory of Heun operators is offered. Some of the topics covered include: quadratic algebras and orthogonal polynomials, differential and difference Heun operators associated to Jacobi and Hahn polynomials, connections with time and band limiting problems in signal processing.

Geoffroy Bergeron, Luc Vinet, Alexei Zhedanov

Some Characterization Problems Related to Sheffer Polynomial Sets

Abstract

In this work, we show some properties of Sheffer polynomials arising from quasi-monomiality. We survey characterization problems dealing with d-orthogonal polynomial sets of Sheffer type. We revisit some families in the literature and we state an explicit formula giving the exact number of Sheffer type d-orthogonal sets. We investigate, in detail, the (d + 1)-fold symmetric case as well as the particular cases d = 1, 2, 3.

Hamza Chaggara, Radhouan Mbarki, Salma Boussorra

From Standard Orthogonal Polynomials to Sobolev Orthogonal Polynomials: The Role of Semiclassical Linear Functionals

Abstract

In this contribution, we present an overview of standard orthogonal polynomials by using an algebraic approach. Discrete Darboux transformations of Jacobi matrices are studied. Next, we emphasize the role of semiclassical orthogonal polynomials as a basic background to analyze sequences of polynomials orthogonal with respect to a Sobolev inner product. Some illustrative examples are discussed. Finally, we summarize some results in multivariate Sobolev orthogonal polynomials.

Juan C. García-Ardila, Francisco Marcellán, Misael E. Marriaga

Two Variable Orthogonal Polynomials and Fejér-Riesz Factorization

Abstract

We consider bivariate polynomials orthogonal on the bicircle with respect to a positive nondegenerate measure. The theory of scalar and matrix orthogonal polynomials is reviewed with an eye toward applying it to the bivariate case. The lexicographical and reverse lexicographical orderings are used to order the monomials for the Gram–Schmidt procedues and recurrence formulas are derived between the polynomials of different degrees. These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the reverse lexicographical ordering. Relations between the coefficients in the recurrence formulas are derived and used to give necessary and sufficient conditions for the existence of a positive measure. These results are then used to construct a class of two variable measures supported on the bicircle that are given by one over the magnitude squared of a stable polynomial. Applications to Fejér–Riesz factorization are also given.

J. S. Geronimo

Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations

Abstract

These are the lecture notes for a course on exceptional polynomials taught at the AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications that took place in Douala (Cameroon) from October 5–12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past 10 years. In addition, some new results are presented on the construction of rational solutions to Painlevé equation P_IV and its higher order generalizations that belong to the $A_{2n}^{(1)}$-Painlevé hierarchy. The construction is based on dressing chains of Schrödinger operators with potentials that are rational extensions of the harmonic oscillator. Some of the material presented here (Sturm-Liouville operators, classical orthogonal polynomials, Darboux-Crum transformations, etc.) are classical and can be found in many textbooks, while some results (genus, interlacing and cyclic Maya diagrams) are new and presented for the first time in this set of lecture notes.

David Gómez-Ullate, Robert Milson

-Rogers–Szegö and Hermite Polynomials, and Induced Deformed Quantum Algebras

Abstract

Deformed quantum algebras, namely the q-deformed algebras and their extensions to (p, q)-deformed algebras, continue to attract much attention. One of the main reasons is that these topics represent a meeting point of nowadays fast developing areas in mathematics and physics like the theory of quantum orthogonal polynomials and special functions, quantum groups, integrable systems, quantum and conformal field theories and statistics.

This contribution paper aims at characterizing the $({\mathcal R},p,q)$-Rogers–Szegö polynomials, and the $({\mathcal R},p,q)$-deformed difference equation giving rise to raising and lowering operators. These polynomials induce some realizations of generalized deformed quantum algebras, (called $({\mathcal R},p,q)$-deformed quantum algebras), which are here explicitly constructed. The study of continuous $({\mathcal R},p,q)$-Hermite polynomials is also performed. Known particular cases are recovered.

Mahouton Norbert Hounkonnou

Zeros of Orthogonal Polynomials

Abstract

In this lecture we discuss properties of zeros of orthogonal polynomials. We review properties that have been used to derive bounds for the zeros of orthogonal polynomials. Topics to be covered include Markov’s theorem on monotonicity of zeros and its generalisations, the proof of a conjecture by Askey and its extensions, interlacing properties of zeros, Sturm’s comparison theorem and convexity of zeros.

Kerstin Jordaan

Properties of Certain Classes of Semiclassical Orthogonal Polynomials

Abstract

In this lecture we discuss properties of orthogonal polynomials for weights which are semiclassical perturbations of classical orthogonality weights. We use the moments, together with the connection between orthogonal polynomials and Painlevé equations to obtain explicit expressions for the recurrence coefficients of polynomials associated with a semiclassical Laguerre and a generalized Freud weight. We analyze the asymptotic behavior of generalized Freud polynomials in two different contexts. We show that unique, positive solutions of the nonlinear difference equation satisfied by the recurrence coefficients exist for all real values of the parameter involved in the semiclassical perturbation but that these solutions are very sensitive to the initial conditions. We prove properties of the zeros of semiclassical Laguerre and generalized Freud polynomials and determine the coefficients a _n,n+j in the differential-difference equation

$$\displaystyle x\frac {d}{dx}P_n(x)=\sum _{k=-1}^{0}a_{n,n+k}P_{n+k}(x), $$

where P _n(x) are the generalized Freud polynomials. Finally, we show that the only monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty }$ that satisfy

$$\displaystyle \pi (x)\mathcal {D}_{q}^2P_{n}(x)=\sum _{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\; x=\cos \theta ,\;~ a_{n,n-2}\neq 0,~ n=2,3,\dots , $$

where π(x) is a polynomial of degree at most 4 and $\mathcal {D}_{q}$ is the Askey–Wilson operator, are Askey–Wilson polynomials and their special or limiting cases, using this relation to derive bounds for the extreme zeros of Askey–Wilson polynomials.

Kerstin Jordaan

Orthogonal Polynomials and Computer Algebra

Abstract

Classical orthogonal polynomials of the Askey–Wilson scheme have extremely many different properties, e.g. satisfying differential equations, recurrence equations, having hypergeometric representations, Rodrigues formulas, generating functions, moment representations etc. Using computer algebra it is possible to switch between one representation and another algorithmically. Such algorithms will be discussed and implementations are presented using Maple.

Wolfram Koepf

Spin Chains, Graphs and State Revival

Abstract

Connections between the 1-excitation dynamics of spin lattices and quantum walks on graphs will be surveyed. Attention will be paid to perfect state transfer (PST) and fractional revival (FR) as well as to the role played by orthogonal polynomials in the study of these phenomena. Included is a discussion of the ordered Hamming scheme, its relation to multivariate Krawtchouk polynomials of the Tratnik type, the exploration of quantum walks on graphs of this association scheme and their projection to spin lattices with PST and FR.

Hiroshi Miki, Satoshi Tsujimoto, Luc Vinet

An Introduction to Special Functions with Some Applications to Quantum Mechanics

Abstract

A short review on special functions and solution of the Coulomb problems in quantum mechanics is given. Multiparameter wave functions of linear harmonic oscillator, which cannot be obtained by the standard separation of variables, are discussed. Expectation values in relativistic Coulomb problems are studied by computer algebra methods.

Sergei K. Suslov, José M. Vega-Guzmán, Kamal Barley

Orthogonal and Multiple Orthogonal Polynomials, Random Matrices, and Painlevé Equations

Abstract

Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics and probability and many other disciplines. In these notes we give an introduction to the use of orthogonal polynomials in random matrix theory, we explain the notion of multiple orthogonal polynomials, and we show the link with certain non-linear difference and differential equations known as Painlevé equations.

Walter Van Assche

Titel: Orthogonal Polynomials
herausgegeben von: Prof. Dr. Mama Foupouagnigni
Wolfram Koepf
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-36744-2
Print ISBN: 978-3-030-36743-5
DOI: https://doi.org/10.1007/978-3-030-36744-2