Jacobi-Type orthogonal polynomials: holonomic equation and electrostatic interpretation (original) (raw)

The Diagonal General Case of the Laguerre-Sobolev Type Orthogonal Polynomials

Revista Colombiana de Matemáticas, 2013

We consider the family of polynomials orthogonal with respect to the Sobolev type inner product corresponding to the diagonal general case of the Laguerre-Sobolev type orthogonal polynomials. We analyze some properties of these polynomials, such as the holonomic equation that they satisfy and, as an application, an electrostatic interpretation of their zeros. We also obtain a representation of such polynomials as a hypergeometric function, and study the behavior of their zeros.

Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros

Journal of Computational and Applied Mathematics, 1998

We obtain an explicit expression for the Sobolev-type orthogonal polynomials Q n (x) associated with the inner product < p; q >= Z 1 ?1 p(x)q(x) (x)dx + A 1 p(1)q(1) + B 1 p(?1)q(?1) + A 2 p 0 (1)q 0 (1) + B 2 p 0 (?1)q 0 (?1); where (x) = (1?x) (1+x) is the Jacobi weight function, ; > ?1, A 1 ; B 1 ; A 2 ; B 2 0 and p, q 2 IP, the linear space of polynomials with real coe cients. The hypergeometric representation (6 F 5) and the second order linear di erential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in-1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Q n (x). Such a zero is located outside the interval-1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.

An electrostatics model for zeros of general orthogonal polynomials

Pacific Journal of Mathematics, 2000

We prove that the zeros of general orthogonal polynomials, subject to certain integrability conditions on their weight functions determine the equilibrium position of movable n unit charges in an external field determined by the weight function. We compute the total energy of the system in terms of the recursion coefficients of the orthonormal polynomials and study its limiting behavior as the number of particles tends to infinity in the case of Freud exponential weights.

On quasi-orthogonal polynomials: Their differential equations, discriminants and electrostatics

Journal of Mathematical Analysis and Applications, 2019

In this paper, we develop a general theory of quasi-orthogonal polynomials. We first derive three-term recurrence relation and second-order differential equations for quasi-orthogonal polynomials. We also give an expression for their discriminants in terms of the recursion coefficients of the corresponding orthogonal polynomials. In addition, we investigate an electrostatic equilibrium problem where the equilibrium position of movable charges is attained at the zeros of the quasi-orthogonal polynomials. The examples of the Freud weight w(x) = e −x 4 +2tx 2 and the Jacobi weight w(x) = (1 − x) α (1 + x) β are discussed in some detail. Finally, we consider the nonlinear orthogonality preserving transformation and related matrix problem.

An electrostatic model for zeros of general orthogonal polynomials

2015

We prove that the zeros of general orthogonal polynomi-als, subject to certain integrability conditions on their weight functions determine the equilibrium position of movable n unit charges in an external field determined by the weight function. We compute the total energy of the system in terms of the recursion coefficients of the orthonormal polynomials and study its limiting behavior as the number of particles tends to infinity in the case of Freud exponential weights. 1. Introduction. Stieltjes [24], [25] considered the following electrostatic model. Fix two charges (α + 1)/2 and (β + 1)/2 at x = 1 and x = −1, respectively, then put n movable unit charges at distinct points in (−1, 1). The question is to determine the equilibrium position of the movable charges when the in-

Generalized Jacobi weights, Christoffel functions, and zeros of orthogonal polynomials

Journal of Approximation Theory, 1992

We study generalized Jacobi weight functions in terms of their (generalized) degree. We obtain sharp lower and upper bounds for the corresponding ChristoEefe'. functions, and for the distance of the consecutive zeros of the corresponding orthogonal polynomials. The novelty of our results is that our constants depend only on the degree of the weight function but not on the weight itse!f. !C 1992 Academx Press. Inc.

Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation

Mathematics

We study the sequence of monic polynomials {Sn}n⩾0, orthogonal with respect to the Jacobi-Sobolev inner product ⟨f,g⟩s=∫−11f(x)g(x)dμα,β(x)+∑j=1N∑k=0djλj,kf(k)(cj)g(k)(cj), where N,dj∈Z+, λj,k⩾0, dμα,β(x)=(1−x)α(1+x)βdx, α,β>−1, and cj∈R∖(−1,1). A connection formula that relates the Sobolev polynomials Sn with the Jacobi polynomials is provided, as well as the ladder differential operators for the sequence {Sn}n⩾0 and a second-order differential equation with a polynomial coefficient that they satisfied. We give sufficient conditions under which the zeros of a wide class of Jacobi-Sobolev polynomials can be interpreted as the solution of an electrostatic equilibrium problem of n unit charges moving in the presence of a logarithmic potential. Several examples are presented to illustrate this interpretation.

2. Review of orthogonal polynomials

De Gruyter eBooks, 2017

Review of orthogonal polynomials 2.1 Introduction Developments and interests in orthogonal polynomials have seen continuous and great progress since their appearance. Orthogonal polynomials are connected with many mathematical, physical, engineering, and computer sciences topics, such as trigonometry, hypergeometric series, special and elliptic functions, continued fractions, interpolation, quantum mechanics, partial differential equations. They are also be found in scattering theory, automatic control, signal analysis, potential theory, approximation theory, and numerical analysis. Orthogonal polynomials are special polynomials that are orthogonal with respect to some special weights allowing them to satisfy some properties that are not generally fulfilled with other polynomials or functions. Such properties have made them wellknown candidates to resolve enormous problems in physics, probability, statistics and other fields. Since their origin in the early 19th century, orthogonal polynomials have formed a somehow classical topic related to Legendre polynomials, Stieltjes' continued fractions, and the work of Gauss, Jacobi, and Christoffel, which has been generalized by Chebyshev, Heine, Szegö, Markov, and others. The most popular orthogonal polynomials are Jacobi, Laguerre, Hermite polynomials, and their special relatives, such as Gegenbauer, Chebyshev, and Legendre polynomials. An extending family has been developed from the work of Wilson, inducing a special set of orthogonal polynomials known by his name, which generalizes the Jacobi class. This new family has given rise to other previously unknown sets of orthogonal polynomials, including Meixner Pollaczek, Hahn, and Askey polynomials. Orthogonal polynomials may also be classified according to the measure applied to define the orthogonality. In this context, we cite the class of discrete orthogonal polynomials that form a special case based on some discrete measure. The most common are Racah polynomials, Hahn polynomials, and their dual class, which in turn include Meixner, Krawtchouk, and Charlier polynomials. Already with the classification of orthogonal polynomials, one can distinguish circular and generally spherical orthogonal polynomials, which consists of some special sets related to measures supported by the circle or the sphere. One well-known class is composed of Rogers-Szegö polynomials on the unit circle and Zernike polynomials, which are related to the unit disk. Orthogonal polynomials, and especially classical ones, can generally be introduced by three principal methods. A first method is based on the Rodrigues formula which consists of introducing orthogonal polynomials as outputs of a derivation.