System Of Orthogonal Polynomials Research Articles

Finite perturbations of orthogonal polynomials

In this paper we study orthogonal polynomials ( p n ) which arise from a given system of orthogonal polynomials ( p n ) by a finite perturbation of the recurrence coefficients, i.e., the recurrence coefficients ( α n ), ( λ n + 1 ) of ( p n ) respectively (α n ), (λ n + 1 ) of ( p n ) are related to each other by α n + l = α n + m and λ n + 1 + l = λ n + 1 + m for Nϵ N , where l and m are fixed nonnegative integers. Closed expressions for the polynomial p n in terms of p n and its associated polynomial p n − 1 (1) as well as for the orthogonality measure μ of ( p n ) in terms of the orthogonality measure μ of ( p n ) are given. In fact the results are presented for the most general case when μ is an arbitrary, not necessarily positive, measure.

Linearization and connection coefficients of orthogonal polynomials

Let {P n } =0/∞ be a system of orthogonal polynomials.Lasser [5] observed that if the linearization coefficients of {P n } =0/∞ are nonnegative then each of theP n (x) is a linear combination of the Tchebyshev polynomials with nonnegative coefficients. The aim of this paper is to give a partial converse to this statement. We also consider the problem of determining when the polynomialsP n can be expressed in terms ofQ n with nonnegative coefficients, where {Q n } =0/∞ is another system of orthogonal polynomials. New proofs of well known theorems are given as well as new results and examples are presented.

On orthogonal polynomials transformed by the QR algorithm

In this paper, we study two equivalent ways of transforming a system of orthogonal polynomials: one is the so-called lifting of the recurrence relation associated with an orthogonal polynomial system (OPS) and the other consists of applying the QR algorithm to the Jacobi matrix of the OPS. These transformations are useful for, among other things, identifying the Sobolev orthogonal polynomials with respect to certain measures, which were previously investigated by Iserles et al. (1991). As a by-product we obtain a new class of modified Lommel polynomials.

Continuous Hahn Polynomials of Differential Operator Argument and Analysis on Riemannian Symmetric Spaces of Constant Curvature

AbstractFor the three types of simply connected Riemannian spaces of constant curvature it is shown that the associated spherical functions can be obtained from the corresponding (zonal) spherical functions by application of a differential operator of the form p(i d/dt), where p belongs to a system of orthogonal polynomials: Gegenbauer polynomials, Hahn polynomials or continuous symmetric Hahn polynomials. We give a group theoretic explanation of this phenomenon and relate the properties of the polynomials p to the properties of the corresponding representation. The method is extended to the case of intertwining functions.

Orthogonal polynomials of two discrete arguments associated with bivariate Pascal distribution

We construct a system of orthogonal polynomials of two discrete arguments associated with the bivariate Pascal distribution. Their generating function is derived and a relationship with previously studied polynomials of several discrete arguments is indicated.

Absolutely continuous measures for systems of orthogonal polynomials with unbounded recurrence coefficients

This paper considers systems of orthogonal polynomials which satisfy a three-term recurrence relation in which the recursion coefficients are unbounded. Conditions are imposed on the coefficient sequences to establish that the corresponding measure of orthogonality is absolutely continuous.

Weighted Markov and Bernstein type inequalities for generalized non-negative polynomials

Weighted Markov and Bernstein type inequalities are established for generalized non-negative polynomials and generalized polynomial weight functions. The novelty of the results lies in the fact that in these estimates only the generalized degrees of the generalized polynomial and the generalized polynomial weight function, respectively, and a multiplicative absolute constant show up. To prove such inequalities was motivated by studying systems of orthogonal polynomials simultaneously, associated with generalized Jacobi weight functions.

Compact perturbations of orthogonal polynomials

We investigate orthogonal polynomials on the real line defined by a reccurence relation for which the recurrence coefficients behave asymptotically like a given system of recurrence coefficients. We give the asymptotic behavior of the orthogonal polynomials (relative to the given comparison system of orthogonal polynomials) and from this we deduce properties of the orthogonality measure

THEORY OF DYNAMIC RESPONSE FUNCTIONS OF PERIODICALLY MODULATED PHYSICAL SYSTEMS

Dynamical properties of models of physical systems distinguished in having two essential length scales are derived. The method used is non-perturbative. It is cast in terms of systems of orthogonal polynomials, whose properties are reviewed together with recent work on the spectrum and algebraic curves associated with periodic difference operators. Response functions observed in experiments, which can probe spatial and temporal features, are calculated for several models of current interest, including spin excitations in a modulated magnet and the vibrations of a modulated harmonically coupled chain of atoms.

Solutions of Correspondence Analysis with Artificial Data of Typical Patterns

Solutions of Correspondence Analysis (CA) are obtained with three artificial two-way contingency tables. Two new concepts are introduced to clarify the structure of the problem. One is dual space representation, which is closely related to the representation proposed by Tenenhaus and Youn. (1985). Another is orthogonal polynomial systems for finite sums. Further, these suggest possible procedures for obtaining solutions of other method of multivariate data analysis.

Empirical bayes estimation of the binomial parameter n

Estimation of the prior distribution of the binomial parameter nbased on a system of orthogonal polynomials, the Poisson-Charlier polynomials, is studied. It is shown that the resulting estimator is mean squared consistent with rate O(N e-1), where Nis the sample size and e> 0 is arbitrarily small.

Hydrodynamic equations for a plasma in the region of the distribution-function plateau

A hydrodynamic theory is presented for plasma particles whose distribution function is close to a plateau in some phase-space region for a finite one-dimensional phase-spase interval. In the framework of the methods of moments, a series expansion of the particle distribution function in an appropriate system of orthogonal (Legendre) polynomials near the state with a plateau is carried out. Deviations from the plateau are due to the presence of a heat flux. The expansion obtained for the distribution function is used to truncate the chain of hydrodynamic equations and to calculate the additional terms arising from the finiteness of the phase interval. The case of one-dimensional Langmuir turbulence is considered as an example. Expressions are presented for the moments of the quasilinear integral of collisions, which describe the variations of the macroscopic characteristics of resonant particles in the respective hydrodynamic equations. Criteria for applicability of the equations obtained are presented. Approximate invariants of motion are obtained in the extreme cases of narrow and broad plateaux.

Dynamic subsoil-coupling between rigid, circular foundations on the halfspace

On the surface of a homogeneous, isotropic and linear-elastic halfspace a number of rigid circular foundations are set to harmonic vibrations. The dynamic behaviour of this group of rigid foundations is being described by means of influence functions. These influence functions are gained from the solution of a system of integral equations describing the mixed-boundary value problem. In order to solve the system of integral equations the stress-distributions acting in the contact area between foundation and halfspace are approximated by a system of orthogonal polynomials. This procedure is suitable for the determination of the dynamic behaviour of a group of foundations as well as for the solution of problems of passive screening if the frequency of the excitation is known.

Some Extensions of Askey-Wilson's Q-Beta Integral and the Corresponding Orthogonal Systems

AbstractA seven-parameter extension of Askey and Wilson's four parameter q-beta integral is written in a symmetric form as the sum of multiples of two very-well-poised balanced basic hypergeometric 10Φ9 series. Two special cases are considered in which the evaluation of the integral gives single terms by the q-Dixon formula in one case and by a special case of the Verma-Jain formula in the other. An orthogonal polynomial system is obtained in the first case and a system of biorthogonal rational function is obtained in the second. It is also shown that the biorthogonal system represents a generalization of Rogers’ q-ultraspherical polynomials.

On two sets of orthogonal polynomial systems encountered in nonlinear physics

Two sets with an infinite number of new systems of orthogonal polynomials have recently been discovered by Smith in connection with some non-linear physical problems, e.g. the dispersion of a buoyant contaminant in a fluid. They appear as solutions of non-linear differential equations. Let {P,(x; m, k, S)} and (Q,(x; m, k)}, with n = 0, k, m + k, 2m, 2m + k, . . ., denote a generic system of each set. The positive integers k and m are restricted by k 1 - k. Although the orthogonality interval of these polynomials is real, their zeros are generally complex. Here the sum rules y, = X x;., r = 1.2,. . ., for the zeros {x ,,,,; i = 1,2,. . . , n} of the nth-degree polynomials of these sets are studied. It is found that all these quantities vanish except for r = pm, p being an arbitrary positive integer. Simple recurrent expressions for ypn, are given.

Systems of orthogonal polynomials explicitly represented by the Jacobi polynomials

Systems of orthogonal polynomials explicitly represented by the Jacobi polynomials

Analytic properties of the recursion method in the presence of band gaps

The recursion method gives a good approximation of the density of states in the single-band case, where the recursion coefficients converge. When band gaps exist, it is known that they exhibit asymptotic undamped oscillations. In order to clarify these asymptotic behaviours, an analytic study of the recursion method applicable to the band-gap case is presented. The asymptotic properties of the system of orthogonal polynomials to which this method is closely related, are investigated by use of the concept of the exterior mapping function. Various relations, which connect the means of the recursion coefficients to the support of a spectrum, are obtained. The period of their asymptotic oscillations is written as a function of the support. The numerical results by Turchi, Ducastelle and Treglia (1982) can be well interpreted by the present theory. The asymptotic properties of the truncated basis are also examined.

A program to set up systems of orthogonal polynomials

A program to set up systems of orthogonal polynomials

Plancherel-Rotach-type asymptotics for orthogonal polynomials associated with [formula omitted

All orthogonal polynomial systems satisfy a recurrence formula: xp n ( x) = a n + 1 p n + 1 ( x) + b n p n ( x) + a n p n − 1( x) . (1) In case the weight function (over (− ∞, ∞)) is exp( −x 6 6 ), b n = 0 , and the a n also satisfy the recurrence formula: n = a n 2( a n + 2 2 a n + 1 2 + a n + 1 4 + 2 a n + 1 2 a n 2 + a n + 1 2 a n 2 + a n 4 + 2 a n 2 a n−1 2 + a n−1 4 + a n−2 2 a n 2). The existence of an asymptotic series for a n has been recently proved by A. Máté and P. G. Nevai. The present paper is based on the above two recurrence formulas. First, we apply Shohat's method to (1) to obtain a differential equation for P n ( x). Then, by applying the asymptotics of a n to this differential equation, we obtain Plancherel-Rotach-type asymptotics for P n ( x) in the interval ¦x¦ ⩽ cx n , where x n denotes the largest zero of P n ( x), and 0 < c < 1.

SOME SYSTEMS OF ORTHOGONAL POLYNOMIALS ASSOCIATED WITH SINGULAR INTEGRAL EQUATIONS

In this paper, some systems of orthogonal polynomials associated with singular integral equation of the form a(t)φ(t)+b(t)π∫-11φ(t)τ-t dτ=f(t), -1<t<1is derived, and some properties of such polynomials will are discussed.