Showing papers on "Discrete orthogonal polynomials published in 2000"

Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach

Abstract: Riemann-Hilbert problems Jacobi operators Orthogonal polynomials Continued fractions Random matrix theory Equilibrium measures Asymptotics for orthogonal polynomials Universality Bibliography.

Stochastic processes and orthogonal polynomials

Abstract: 1 The Askey Scheme of Orthogonal Polynomials.- 2.1 Markov Processes.- 3 Birth and Death Processes, Random Walks, and Orthogonal Polynomials.- 4 Sheffer Systems.- 5 Orthogonal Polynomials in Stochastic Integration Theory.- Stein Approximation and Orthogonal Polynomials.- Conclusion.- A Distributions.- B Tables of Classical Orthogonal Polynomials.- B.1 Hermite Polynomials and the Normal Distribution.- B.2 Scaled Hermite Polynomials and the Standard Normal Distribution.- B.3 Hermite Polynomials with Parameter and the Normal Distribution.- B.4 Charlier Polynomials and the Poisson Distribution.- B.5 Laguerre Polynomials and the Gamma Distribution.- B.6 Meixner Polynomials and the Pascal Distribution.- B.7 Krawtchouk Polynomials and the Binomial Distribution.- B.8 Jacobi Polynomials and the Beta Kernel.- B.9 Hahn Polynomials and the Hypergeometric Distribution.- C Table of Duality Relations Between Classical Orthogonal Polynomials.- D Tables of Sheffer Systems.- D.1 Sheffer Polynomials and Their Generating Functions.- D.2 Sheffer Polynomials and Their Associated Distributions.- D.3 Martingale Relations with Sheffer Polynomials.- E Tables of Limit Relations Between Orthogonal Polynomials in the Askey Scheme.- References.

Polynomials with Special Regard to Reducibility

Abstract: This book covers most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields and finitely generated fields. Results valid only over finite fields, local fields or the rational field are not covered here, but several theorems on reducibility of polynomials over number fields that are either totally real or complex multiplication fields are included. Some of these results are based on recent work of E. Bombieri and U. Zannier (presented here by Zannier in an appendix). The book also treats other subjects like Ritt's theory of composition of polynomials, and properties of the Mahler measure, and it concludes with a bibliography of over 300 items. This unique work will be a necessary resource for all number theorists and researchers in related fields.

Combinatorial Aspects of the K-Theory of Grassmannians

Abstract: In this paper we study Grothendieck polynomials indexed by Grassmannian permutations, which are representatives for the classes corresponding to the structure sheaves of Schubert varieties in the K-theory of Grassmannians. These Grothendieck polynomials are nonhomogeneous symmetric polynomials whose lowest homogeneous component is a Schur polynomial. Our treatment, which is closely related to the theory of Schur functions, gives new information about these polynomials. Our main results are concerned with the transition matrices between Grothendieck polynomials indexed by Grassmannian permutations and Schur polynomials on the one hand and a Pieri formula for these Grothendieck polynomials on the other.

An electrostatics model for zeros of general orthogonal polynomials

Abstract: 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 the Connection Between Macdonald Polynomials and Demazure Characters

Abstract: We show that the specialization of nonsymmetric Macdonald polynomials at t e 0 are, up to multiplication by a simple factor, characters of Demazure modules for \widehat{sl(n)}. This connection furnishes Lie-theoretic proofs of the nonnegativity and monotonicity of Kostka polynomials.

General Hermite and Laguerre two-dimensional polynomials

Abstract: General Hermite and Laguerre two-dimensional (2D) polynomials which form a (complex) three-parameter unification of the special Hermite and Laguerre 2D polynomials are defined and investigated. The general Hermite 2D polynomials are related to the two-variable Hermite polynomials but are not the same. The advantage of the newly introduced Hermite and Laguerre 2D polynomials is that they satisfy orthogonality relations in a direct way, whereas for the purpose of orthonormalization of the two-variable Hermite polynomials two different sets of such polynomials are introduced which are biorthonormal to each other. The matrix which plays a role in the new definition of Hermite and Laguerre 2D polynomials is in a considered sense the square root of the matrix which plays a role in the definition of two-variable Hermite polynomials. Two essentially different explicit representations of the Hermite and Laguerre 2D polynomials are derived where the first involves Jacobi polynomials as coefficients in superpositions of special Hermite or Laguerre 2D polynomials and the second is a superposition of products of two Hermite polynomials with decreasing indices and with coefficients related to the special Laguerre 2D polynomials. Generating functions are derived for the Hermite and Laguerre 2D polynomials.

Multiple orthogonal polynomials associated with macdonald functions

Abstract: We consider multiple orthogonal polynomials corresponding to two Macdonald functions (modified Bassel functions of the second kind), with emphasis on the polynomials on the diagonal of the Hermite-Pade table. We give some properties of these polynomials: differential properties, a Rodrigues type formula and explicit formulas for the third order linear recurrence relation.

On the classical {$d$}-orthogonal polynomials defined by certain generating functions. II

Abstract: The purpose of this work is to present some results on the d-orthogonal polynomials dened by generating functions of certain forms to be specied below. The resulting polynomials are natural extensions of some classical orthogonal polynomials. The rst part of this study is motived by the recent work of Von Bachhaus [21] who showed that, among the orthogonal polynomials, only the Hermite and the Gegenbauer polynomials are dened by the generating function G 2xt t 2 . Here we generalize this result in the context of d-orthogonality, by considering the polynomials generated by G (d +1)xt t d+1 ,w hered is a positive integer. We obtain that the resulting polynomials are d-symmetric Denition 1.2 and \classical" in the Hahn’s sense. We provide some examples to illustrate the results obtained and show that they involve certain known polynomials. Finally, we conclude by giving some properties of the zeros of these polynomials as well as a (d +1 )-order dierential equation satised by each polynomial. In forthcoming paper [2] we will consider the polynomials generated by e t (xt).

Computing Roots of Polynomials over Function Fields of Curves

Abstract: We design algorithms for finding roots of polynomials over function fields of curves. Such algorithms are useful for list decoding of Reed-Solomon and algebraic-geometric codes. In the first half of the paper we will focus on bivariate polynomials, i.e., polynomials over the coordinate ring of the affine line. In the second half we will design algorithms for computing roots of polynomials over the function field of a nonsingular absolutely irreducible plane algebraic curve. Several examples are included.

Information entropy of Gegenbauer polynomials

Abstract: The information entropy of Gegenbauer polynomials is relevant since this is related to the angular part of the information entropies of certain quantum mechanical systems such as the harmonic oscillator and the hydrogen atom in D dimensions. We give an effective method to compute the entropy for Gegenbauer polynomials with an integer parameter and obtain the first few terms in the asymptotic expansion as the degree of the polynomial tends to infinity.

Robust arithmetic for multivariate Bernstein-form polynomials

Abstract: There are several ways to represent, to handle and to display curved surfaces in computer-aided geometric design that involve the use of polynomials. This paper deals with polynomials in the Bernstein form. Other work has shown that these polynomials are more numerically stable and robust than power-form polynomials. However, these advantages are lost if conversions to and from the customary power form are made. To avoid this, algebraic manipulations have to be done in the Bernstein basis. Farouki and Rajan (R.T. Farouki, V.T. Rajan, Algorithms for polynomials in Bernstein form, Computer Aided Geometric Design 5 (1988) 1–26) present methods for doing arithmetic on univariate Bernstein-basis polynomials. This paper extends all polynomial arithmetic operations to multivariate Bernstein-form polynomials.

Convex optimization over positive polynomials and filter design

Abstract: Positive polynomial matrices play a fundamental role in systems and control theory: they represent e.g. spectral density functions of stochastic processes and show up in spectral factorizations, robust control and filter design problems. Positive polynomials obviously form a convex set and were recently studied in the area of convex optimization [1, 5]. It was shown in [2, 5] that positive polynomial matrices can be parametrized using block Hankel and Toeplitz matrices. In this paper, we use this parametrization to derive efficient computational algorithms for optimization problems over positive polynomials. Moreover, we show that filter design problems can be solved using these results. Keywords: convex optimization, positive polynomials, trigonometric polynomials, filter design.

More on Electrostatic Models for Zeros of Orthogonal Polynomials

Abstract: We consider polynomials orthagonal with respect to a measure μ with an absolutely continuous component and a finite discrete part. We prove that subject to certatin integrability conditions, the polynomials satisfy a second order differential equation. The zeroes of such polynomials determine the equilibrium position of movable n unit charges in an external field determined by the measure μ. We also evaluate the discriminant of such orthagonal polynomials and use it to compute the total energy of the system at equilibrium in terms of the recursion coefficients of the orthonormal polynomials. We also investigate several explicit models, the Koornwinder polynomials, the Ginzburg-Landau potential and the generalized Jacobi weights.

Tensor product of principal unitary representations of quantum Lorentz group and Askey–Wilson polynomials

Abstract: We study the tensor product of principal unitary representations of the quantum Lorentz group, prove a decomposition theorem, and compute the associated intertwiners. We show that these intertwiners can be expressed in terms of complex continuations of 6j symbols of Uq(su(2)). These intertwiners are expressed in terms of q-Racah polynomials and Askey–Wilson polynomials. The orthogonality of these intertwiners imply some relation mixing these two families of polynomials. The simplest of these relations is the orthogonality of Askey–Wilson polynomials.

A generalized hypergeometric d -orthogonal polynomial set

Abstract: In this Note, we present an extension of Abdul-Halim and Al-Salam's result [1] in the context of d -orthogonality. The resulting polynomials are analogous to the classical Laguerre polynomials. We provide some of their properties.

Strong Asymptotics of Orthonormal Polynomials with the Aid of Green's Function

Abstract: We give explicit asymptotic representations as well as ratio asymptotics of the orthogonal polynomials with asymptotically periodic reflection coefficients in terms of Green's function. All the limit relations will hold uniformly compact outside the support of the measure of orthogonality. Furthermore, with the help of harmonic measures we will characterize all those sets, i.e., supports of orthogonality measures, where orthogonal polynomials with asymptotically periodic reflection coefficients exist.

Rudin-Shapiro-like polynomials in L 4

Abstract: We examine sequences of polynomials with {+1, -1} coefficients constructed using the iterations p(x) → p(x) ± x d+1 p * (-x), where d is the degree of p and p * is the reciprocal polynomial of p. If po = 1 these generate the Rudin-Shapiro polynomials, We show that the L4 norm of these polynomials is explicitly computable. We are particularly interested in the case where the iteration produces sequences with smallest possible asymptotic L4 norm (or, equivalently, with largest possible asymptotic merit factor). The Rudin-Shapiro polynomials form one such sequence. We determine all p 0 of degree less than 40 that generate sequences under the iteration with this property. These sequences have asymptotic merit factor 3. The first really distinct example has a p 0 of degree 19.

Factoring polynomials over p-adic fields

Abstract: We give an efficient algorithm for factoring polynomials over finite algebraic extensions of the p-adic numbers. This algorithm uses ideas of Chistov's random polynomial-time algorithm, and is suitable for practical implementation.

Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities

Abstract: We briefly review some asymptotics of orthonormal polynomials Then we derive the Bernstein–Szegő, the Riemann–Hilbert (or Fokas–Its–Kitaev), and Rakhmanov projection identities for orthogonal polynomials and attempt a comparison of their applications in asymptotics

Bivariate factorizations connecting Dickson polynomials and Galois theory

Abstract: In his Ph.D. Thesis of 1897, Dickson introduced certain permutation polynomials whose Galois groups are essentially the dihedral groups. These are now called Dickson polynomials of the first kind, to distinguish them from their variations introduced by Schur in 1923, which are now called Dickson polynomials of the second kind. In the last few decades there have been extensive investigations of both of these types, which are related to the classical Chebyshev polynomials. We give new bivariate factorizations involving both types of Dickson polynomials. These factorizations demonstrate certain isomorphisms between dihedral groups and orthogonal groups, and lead to the construction of explicit equations with orthogonal groups as Galois groups.

Some results for a class of generalized polynomials

Abstract: A class of generalized polynomials is considered consisting of the null spaces of certain differential operators with constant coefficients. This class strictly contains ordinary polynomials and appropriately scaled trigonometric polynomials. An analog of the classical Bernstein operator is introduced and it is shown that generalized Bernstein polynomials of a continuous function converge to this function. A convergence result is also proved for degree elevation of the generalized polynomials. Moreover, the geometric nature of these functions is discussed and a connection with certain rational parametric curves is established.

Examples of T-spaces with an?infinite basis

Abstract: This work is devoted to the construction of T-spaces with an infinite basis over a field of finite characteristic and over some other rings. Examples of T-spaces are given that are generated by polynomials in two variables or by polynomials of bounded degree in each variable.