Polynomials whose coefficients coincide with their zeros

Companion orthogonal polynomials: some applications

A general methodology is developed for approximating the distribution of a random variable on the basis of its exact moments. More specifically, a probability density function is approximated by the product of a suitable weight function and a linear combination of its associated orthogonal polynomials. A technique for generating a sequence of orthogonal polynomials from a given weight function is provided and the coefficients of the linear combination are explicitly expressed in terms of the moments of the target distribution. On applying this approach to several test statistics, we observed that the resulting percentiles are consistently in excellent agreement with the tabulated values. As well, it is explained that the same moment-matching technique can be utilized to produce density estimates on the basis of the sample moments obtained from a given set of observations. An example involving a well-known data set illustrates the density estimation methodology advocated herein.

On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials

The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties\n\t\t\t\t as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional U\n\t\t\t\t U = Jα,β + A1δ(x − 1) + B1δ(x + 1) − A2δ \n\t\t\t\t (x − 1) − B2δ \n\t\t\t\t (x + 1),\n\t\t\t\t where Jα,β is the Jacobi linear functional, i.e.\n\t\t\t\t Jα,β , p = 1\n\t\t\t\t −1\n\t\t\t\t p(x)(1 − x)α(1 + x)β dx, α, β > −1, p ∈ P,\n\t\t\t\t and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (−1, 1) (inner asymptotics) and C \\ [−1, 1] (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional U is a generalization of one studied by T. H. Koornwinder when A2 = B2 = 0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi–Markov\n\t\t\t\t function by a rational function with two double poles at ±1. The denominators of the [n−1/n] Padé approximants are our orthogonal polynomials.

In this paper, we study necessary and sufficient conditions for the relation $$\begin{array}{@{}l}P_n^{{[r]}}(x) + a_{n-1,r} P_{n-1}^{{[r]}}(x)= R_{n-r}(x) + b_{n-1,r} R_{n-r-1}(x),\\[5pt]\quad a_{n-1,r}\neq0,\ n\geq r+1,\end{array}$$ where {P n (x)} n?0 and {R n (x)} n?0 are two sequences of monic orthogonal polynomials with respect to the quasi-definite linear functionals $\mathcal{U},\mathcal{V}$ , respectively, or associated with two positive Borel measures μ 0,μ 1 supported on the real line. We deduce the connection with Sobolev orthogonal polynomials, the relations between these functionals as well as their corresponding formal Stieltjes series. As sake of example, we find the coherent pairs when one of the linear functionals is classical.

In this paper, we consider two questions about topological entropy of dynamical systems. We propose to resolve these questions by the same approach of using étale analogs of topological and algebraic dynamical systems. The first question is to define topological entropy for a topological dynamical system ( f , X , Ω ) (f,X,\Omega ) . The main idea is to make use of, in addition to invariant compact subspaces of ( X , Ω ) (X,\Omega ) , compactifications of étale covers π : ( f ′ , X ′ , Ω ′ ) → ( f , X , Ω ) \pi :(f’,X’,\Omega ’)\rightarrow (f,X,\Omega ) ; that is, π ∘ f ′ = f ∘ π \pi \circ f’=f\circ \pi and the fibers of π \pi are all finite. We prove some basic results and propose a conjecture, whose validity allows us to prove further results. The second question is to define topological entropy for algebraic dynamical systems, with the requirement that it should be as close to the pullback on cohomology groups as possible. To this end, we develop an étale analog of algebraic dynamical systems.

The boundary quotient for algebraic dynamical systems

Orthogonality of linear combinations of two orthogonal polynomial sequences

Numerical computation of BCOPs in two variables for solving the vibration problem of a CF-elliptical plate

In this paper, we determine all pairs of orthogonal polynomial sequences $\{ p_n (x)\} $ and $\{ q_n (x)\} $, such that their convolution, \[ Q_n (x,y) = \sum_{k = 0}^n {p_k (x)q_{n - k} (y)} ,\quad n \geqq 0.\] defines $\{ Q_n (x,y)\} $ as an orthogonal polynomial sequence in x for all y. All such triples are determined explicitly in terms of their three-term recurrence formulas. Generating functions and “explicit” representation formulas are obtained. The resulting sequences are found to consist of a class of orthogonal polynomials characterized by J. Meixner (which class includes the Laguerre and Hermite polynomials) together with a new class of orthogonal polynomials which includes the orthogonal q-polynomials of Al-Salam and Carlitz. Explicit orthogonality relations are found for one new special case of this latter class.

A given weight function \(p(x)\) on an interval \([a, b]\) defines uniquely, subject to normalization, a sequence of orthogonal polynomials \(P_n(x)\) and their corresponding sequence of functions of the second kind $$\begin{aligned} Q_n(x)=\int _a^b\frac{P_n(t)p(t)dt}{x-t},\quad \ x\in \mathbf{C } \backslash [a,b]. \end{aligned}$$ corresponding to them. This paper focuses on the reverse problem of characterizing orthogonal polynomials by means of functions of the second kind, together with the properties of such functions of the second kind. Functions of the second kind for orthogonal polynomials are also of particular interest in that they differ only slightly from the second solution of the differential equation satisfied by the orthogonal polynomials.

2 × 2 hypergeometric operators with diagonal eigenvalues

In this chapter, we address several of the many applications of the classical orthogonal polynomial sequences. These applications include first-order differential equations that characterize linear generating functions, additional first-order differential equations, second-order differential equations (with applications to quantum mechanics), difference equations and numerical integration (Gaussian Quadrature). We first develop each of these applications in a general context and then cover examples using specific Sheffer Sequences, i.e. the Laguerre, Hermite, Charlier, Meixner, Meixner–Pollaczek, and Krawtchouk polynomials.

Let (P ν) be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, let (Ωυ) the monic associated polynomials of (P v), and letA andB be self-reciprocal polynomials. We show that the sequence of polynomials (APυλ+BΩυλ)/Aλ, λ stuitably determined, is a sequence of orthogonal polynomials having, up to a multiplicative complex constant, the same recurrence coefficients as theP ν's from a certain index value onward, and determine the orthogonality measure explicity. Conversely, it is also shown that every sequence of orthogonal polynomials on the unit circle having the same recurrence coefficients from a certain index value onward is of the above form. With the help of these results an explicit representation of the associated polynomials of arbitrary order ofP ν and of the corresponding orthogonality measure and Szego function is obtained. The asymptotic behavior of the associated polynomials is also studied. Finally necessary and suficient conditions are given such that the measure to which the above introduced polynomials are orthogonal is positive.

The paper introduces a new method for finding optimal control of algebraic dynamic systems. The structure of algebraic dynamical systems is nonlinear with quadratic and bilinear terms. A new hybrid extended Fourier series is introduced, and state and control variables of the system are expanded by this series. Moreover, properties of new series are presented, and integration and product operational matrices are obtained. Using operational matrices, optimal control of the systems is converted to a set of simultaneous nonlinear algebraic relations. An illustrative example is included to compare our results with those in the literature.