On Connection, Linearization and Duplication Coefficients of Classical Orthogonal Polynomials

In this work, we have mainly achieved the following: 1. we provide a review of the main methods used for the computation of the connection and linearization coefficients between orthogonal polynomials of a continuous variable, moreover using a new approach, the duplication problem of these polynomial families is solved; 2. we review the main methods used for the computation of the connection and linearization coefficients of orthogonal polynomials of a discrete variable, we solve the duplication and linearization problem of all orthogonal polynomials of a discrete variable; 3. we propose a method to generate the connection, linearization and duplication coefficients for q-orthogonal polynomials; 4. we propose a unified method to obtain these coefficients in a generic way for orthogonal polynomials on quadratic and q-quadratic lattices. Our algorithmic approach to compute linearization, connection and duplication coefficients is based on the one used by Koepf and Schmersau and on the NaViMa algorithm. Our main technique is to use explicit formulas for structural identities of classical orthogonal polynomial systems. We find our results by an application of computer algebra. The major algorithmic tools for our development are Zeilberger’s algorithm, q-Zeilberger’s algorithm, the Petkovšek-van-Hoeij algorithm, the q-Petkovšek-van-Hoeij algorithm, and Algorithm 2.2, p. 20 of Koepf's book "Hypergeometric Summation" and it q-analogue.

Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM

We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a delta-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on delta. We apply the obtained estimates to show exponential convergence with rate O(exp(−b square root N)), N being the number of degrees of freedom and b>0, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate O(exp(−b cubic root N )), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.

L-orthogonal polynomials associated with related measures

A positive measure psi defined on [a, b] such that its moments mu(n) = integral(b)(a)t(n) d psi(t) exist for n = 0, +/-1, +/-2. can be called a strong positive measure on [a, b] When 0 <= a < b <= infinity the sequence of polynomials {Q(n)} defined by integral(b)(a) t(-n+s) Q(n)(t) d psi(t) = 0, s = 0, ., n - 1, exist and they are referred here as L-orthogonal polynomials We look at the connection between two sequences of L-orthogonal polynomials {Q(n)((1))} and {Q(n)((0))} associated with two closely related strong positive measures and th defined on [a, b]. To be precise, the measures are related to each other by (t - kappa) d psi(1)(t) = gamma d psi(0)(t). where (t - kappa)/gamma is positive when t is an element of (n, 6). As applications of our study. numerical generation of new L-orthogonal polynomials and monotonicity properties of the zeros of a certain class of L-orthogonal polynomials are looked at. (C) 2010 IMACS Published by Elsevier B V All rights reserved

A realization of the q-deformed harmonic oscillator: rogers-Szegö and Stieltjes-Wigert polynomials

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

On linear combinations of L-orthogonal polynomials associated with distributions belonging to symmetric classes

This paper deals with the classes S-3(omega, beta, b) of strong distribution functions defined on the interval [beta(2)/b, b], 0 < beta < b <= infinity, where 2 omega epsilon Z. The classification is such that the distribution function psi epsilon S-3(omega, beta, b) has a (reciprocal) symmetry, depending on omega, about the point beta. We consider properties of the L-orthogonal polynomials associated with psi epsilon S-3(omega, beta, b). Through linear combination of these polynomials we relate them to the L-orthogonal polynomials associated with some omega epsilon S-3(1/2, beta, b). (c) 2004 Elsevier B.V. All rights reserved.

Blumenthal's theorem for Laurent orthogonal polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form B-n(x) = (x - beta(n))beta(n-1)(x) - alpha(n)xB(n-2)(x), with positive recurrence coefficients alpha(n+1),beta(n) (n = 1, 2,...). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where alpha(n) --> alpha and beta(n) --> beta and show that the zeros of beta(n) are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials. (C) 2002 Elsevier B.V. (USA).

Lamé differential equations and electrostatics

The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation A(x)y″ + 2B(x)y′ + C(x)y = 0, where A(x),B(x) and C(x) are polynomials of degree p + 1,p and p - 1, is under discussion. We concentrate on the case when A(x) has only real zeros aj and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients rj in the partial fraction decomposition B(x)/A(x) = ∑j p=0 rj/(x - aj), we allow the presence of both positive and negative coefficients rj. The corresponding electrostatic interpretation of the zeros of the solution y(x) as points of equilibrium in an electrostatic field generated by charges rj at aj is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges. © 2000 American Mathematical Society.

Concerning the stability of BDF methods

We present an analysis of A0-stability of BDF methods and proof that zero-stable BDF methods are A0-stable using the Schur-Cohn criterion. With this result we have that zero-stable BDF methods are stiffly-stable. © 2008 American Institute of Physics.

Perturbations on the antidiagonals of Hankel matrices

Given a strongly regular Hankel matrix, and its associated sequence of moments which defines a quasi-definite moment linear functional, we study the perturbation of a fixed moment, i.e., a perturbation of one antidiagonal of the Hankel matrix. We define a linear functional whose action results in such a perturbation and establish necessary and sufficient conditions in order to preserve the quasi-definite character. A relation between the corresponding sequences of orthogonal polynomials is obtained, as well as the asymptotic behavior of their zeros. We also study the invariance of the Laguerre-Hahn class of linear functionals under such perturbation, and determine its relation with the so-called canonical linear spectral transformations. © 2013 Elsevier Ltd. All rights reserved.

Limitantes para os zeros de polinômios gerados por uma relação de recorrência de três termos

Pós-graduação em Matemática - IBILCE

Teorema de Sturm e zeros de polinômios ortogonais

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Polinômios algébricos e trigonométricos com zeros reais

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Monotonicidade de zeros de polinômios ortogonais clássicos

Pós-graduação em Matematica Aplicada e Computacional - FCT

Massless scattering at special kinematics as Jacobi polynomials

We study the scattering equations recently proposed by Cachazo, He and Yuan in the special kinematics where their solutions can be identified with the zeros of the Jacobi polynomials. This allows for a non-trivial two-parameter family of kinematics. We present explicit and compact formulas for the n-gluon and n-graviton partial scattering amplitudes for our special kinematics in terms of Jacobi polynomials. We also provide alternative expressions in terms of gamma functions. We give an interpretation of the common reduced determinant appearing in the amplitudes as the product of the squares of the eigenfrequencies of small oscillations of a system whose equilibrium is the solutions of the scattering equations.

Métodos numéricos para encontrar zeros de funções: aplicações para o Ensino Médio

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)