Showing papers on "Discrete orthogonal polynomials published in 1998"
TL;DR: In this paper, a survey of the application of multiple orthogonal polynomials and analytic theory of two model families of general MOPs, referred to as Angelesco and Nikishin systems, is presented.
228 citations
Book•
01 Jun 1998193 citations
TL;DR: In this article, it was shown that all classical orthogonal polynomials from the Meixner class, when evaluated at a corresponding Levy process, are martingales.
Abstract: We study an unusual connection between orthogonal polynomials and martingales. We prove that all classical orthogonal polynomials from the Meixner class, when evaluated at a corresponding Levy process, are martingales. This result is well known for the case of Hermite polynomials evaluated in Brownian motion. Our results provide similar analogues for the Poisson process, for the Gamma process and for two less familiar processes related to Meixner polynomials
185 citations
TL;DR: In this article, the authors derived convolution identities for Al-Salam and Chihara polynomials by using the Clebsch-Gordan decomposition and the Racah coefficients.
Abstract: Theinterpretation of the Meixner--Pollaczek, Meixner, and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra ${\frak{su}}(1,1)$ and the Clebsch--Gordan decomposition lead to generalizations of the convolution identities for these polynomials. Using the Racah coefficients, convolution identities for continuous Hahn, Hahn, and Jacobi polynomials are obtained. From the quantized universal enveloping algebra for ${\frak{su}}(1,1)$, convolution identities for the Al-Salam and Chihara polynomials and the Askey--Wilson polynomials are derived by using the Clebsch--Gordan and Racah coefficients. For the quantized universal enveloping algebra for ${\frak{su}}(2)$, q-Racah polynomials are interpreted as Clebsch--Gordan coefficients, and the linearization coefficients for a two-parameter family of Askey--Wilson polynomials are derived.
135 citations
TL;DR: Two algorithms are presented, both rely on the expansion of a multivariate polynomial into Bernstein polynomials and the first is an improvement of the so-called Bernstein algorithm and checks the Hurwitz determinant for positivity over the parameter set.
Abstract: This paper considers the robust stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals. Two algorithms are presented, both rely on the expansion of a multivariate polynomial into Bernstein polynomials. The first one is an improvement of the so-called Bernstein algorithm and checks the Hurwitz determinant for positivity over the parameter set. The second one is based on the analysis of the value set of the family of polynomials and profits from the convex hull property of the Bernstein polynomials. Numerical results to real-world control problems are presented showing the efficiency of both algorithms.
127 citations
Book•
15 Aug 1998
TL;DR: In this paper, level crossing of stochastic processes is considered for algebraic and trigonometric polynomials, and complex coefficients and complex roots are considered for complex linear polynomials.
Abstract: Preface. Introduction. Level Crossings of Stochastic Processes. Algebraic Polynomials. Trigonometric Polynomials. Orthogonal Polynomials. Hyperbolic Polynomials. Other Distributions. Complex Coefficients and Complex Roots. Bibliography.
123 citations
TL;DR: In this paper, the Hermite matrix polynomials are used to approximate matrix functions appearing in the solution of differential systems in terms of the three-term recurrence formula.
100 citations
TL;DR: In this paper, an electrostatic interpretation of the zeros of a large class of orthogonal polynomials is presented, based on the ideas of Stieltjes.
67 citations
TL;DR: This paper provides a numeric parameter for determining whether two polynomials are prime, even under small perturbations of the coefficients, and relies on an inversion formula for Sylvester matrices to establish an effective criterion for relative primeness.
65 citations
TL;DR: In this paper, a complete basis of eigenfunctions of Yamamoto's BN spin Calogero model is obtained by multiplying these polynomials by the ground state.
Abstract: There are examples of Calogero–Sutherland models associated to the Weyl groups of type A and B. When exchange terms are added to the Hamiltonians the systems have non-symmetric eigenfunctions, which can be expressed as products of the ground state with members of a family of orthogonal polynomials. These polynomials can be defined and studied by using the differential-difference operators introduced by the author in Trans. Am. Math. Soc. 311, 167–183 (1989). After a description of known results, particularly from the works of Baker and Forrester, and Sahi; there is a study of polynomials which are invariant or alternating for parabolic subgroups of the symmetric group. The detailed analysis depends on using two bases of polynomials, one of which transforms monomially under group actions and the other one is orthogonal. There are formulas for norms and point-evaluations which are simplifications of those of Sahi. For any parabolic subgroup of the symmetric group there is a skew operator on polynomials which leads to evaluation at (1,1,… ,1) of the quotient of the unique skew polynomial in a given irreducible subspace by the minimum alternating polynomial, analogously to a Weyl character formula. The last section concerns orthogonal polynomials for the type B Weyl group with an emphasis on the Hermite-type polynomials. These can be expressed by using the generalized binomial coefficients. A complete basis of eigenfunctions of Yamamoto's BN spin Calogero model is obtained by multiplying these polynomials by the ground state.
63 citations
TL;DR: In this article, the authors give an overview of recent work on the distribution of zeros of discrete orthogonal polynomials and give a classification of polynomial zeros.
TL;DR: In this paper, the authors considered the Calogero-Sutherland model with n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 28) and constructed an isomorphism from the space of states into a space of symmetric Laurent polynomials that maps the eigenbasis into the basis of gl N - Jack polynomorphisms.
Abstract: We consider the gl N -invariant Calogero—Sutherland models with N = 1, 2, 3, … in the framework of symmetric polynomials. The Hamiltonian of any such model admits a distinguished orthogonal eigenbasis characterized as the union of Yangian Gelfand—Zetlin bases of irreducible components with respect to the Yangian action on the space of states. We construct an isomorphism from the space of states into the space of symmetric Laurent polynomials that maps the eigenbasis into the basis of gl N - Jack polynomials. These polynomials are defined as specializations of Macdonald polynomials where both parameters approach an Nth primitive root of unity. As an application of this isomorphism we compute two-point dynamical spin-density and density correlation functions in the gl 2-invariant Calogero-Sutherland model at integer values of the coupling constant.
TL;DR: In this article, it was shown that a large class of cubature formulae on the unit sphere can be derived from those on a unit ball and vice versa, and the results provided a new approach to study orthogonal polynomials and cubatures on spheres.
Abstract: Orthogonal polynomials on the unit sphere in RRd+1 and on the unit ball in RRd are shown to be closely related to each other for symmetric weight functions. Furthermore, it is shown that a large class of cubature formulae on the unit sphere can be derived from those on the unit ball and vice versa. The results provide a new approach to study orthogonal polynomials and cubature formulae on spheres.
TL;DR: In this paper, the Sobolev orthogonal polynomials are defined by a signed Borel measure on the real line with finite moments of all orders, where each moment corresponds to a Borel metric.
Abstract: Recently many people have studied the Sobolev orthogonal polynomials, that is, polynomials which are orthogonal relative to a symmetric bilinear form $\\phi(\\cdot,\\cdot)$ defined by $$ (1.1) $\\phi(p,q) := (p,q)_N = \\sum_{k=0}^{N} \\int_{R}p^(k) (x)q^(k) (x) d\\mu_k, $$ where each $d\\mu_k$ is a signed Borel measure on the real line $R$ with finite moments of all orders. For the brief history on this subject, we refer to the survey article Ronveaux [13] and Marcellan and et al [10].
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TL;DR: In this article, the authors obtained new combinatorial formulae for modified Hall-Littlewood polynomials, for matrix elements of the transition matrix between the elementary symmetric polynomial and Hall-littlewood's ones, and for the number of rational points over the finite field of unipotent partial flag variety.
Abstract: We obtain new combinatorial formulae for modified Hall–Littlewood polynomials, for matrix elements of the transition matrix between the elementary symmetric polynomials and Hall-Littlewood’s ones, and for the number of rational points over the finite field of unipotent partial flag variety. The definitions and examples of generalized mahonian statistic on the set of transport matrices and dual mahonian statistic on the set of transport (0,1)–matrices are given. We also review known q–analogues of Littlewood– Richardson numbers and consider their possible generalizations. Some conjectures about multinomial fermionic formulae for homogeneous unrestricted one dimensional sums and generalized Kostka–Foulkes polynomials are formulated. Finally we suggest two parameter deformations of polynomials P�µ(t) and one dimensional sums.
TL;DR: In this article, positive discrete series representations of the Lie algebra su(1, 1) and the quantum algebra Uq(su( 1, 1)) are considered and the diagonalization of a self-adjoint operator in these representations and in tensor products of such representations is determined, and the generalized eigenvectors are constructed in terms of orthogonal polynomials.
Abstract: Positive discrete series representations of the Lie algebra su(1,1) and the quantum algebra Uq(su(1,1)) are considered. The diagonalization of a self-adjoint operator (the Hamiltonian) in these representations and in tensor products of such representations is determined, and the generalized eigenvectors are constructed in terms of orthogonal polynomials. Using simple realizations of su(1,1), Uq(su(1,1)), and their representations, these generalized eigenvectors are shown to coincide with generating functions for orthogonal polynomials. The relations valid in the tensor product representations then give rise to new generating functions for orthogonal polynomials, or to Poisson kernels. In particular, a group theoretical derivation of the Poisson kernel for Meixner–Pollaczek and Al-Salam–Chihara polynomials is obtained.
TL;DR: In this article, the authors studied the discrete orthogonality measure of the Askey-Wilson polynomials with parameters satisfying a truncation condition such that the orthogonal measure becomes discrete with support on a finite grid.
Abstract: The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials is studied for parameters satisfying a truncation condition such that the orthogonality measure becomes discrete with support on a finite grid. For this parameter regime the polynomials may be seen as a multivariable counterpart of the (one-variable) $q$-Racah polynomials. We present the discrete orthogonality measure, expressions for the normalization constants converting the polynomials into an orthonormal system (in terms of the normalization constant for the unit polynomial), and we discuss the limit $q\\to 1$ leading to multivariable Racah type polynomials. Of special interest is the situation that $q$ lies on the unit circle; in that case it is found that there exists a natural parameter domain for which the discrete orthogonality measure (which is complex in general) becomes real-valued and positive. We investigate the properties of a finite-dimensional discrete integral transform for functions over the grid, whose kernel is determined by the multivariable $q$-Racah polynomials with parameters in this positivity domain.
TL;DR: In this paper, the strong asymptotics for the class of Krawtchouk polynomials on the real line were derived and shown to describe the limiting distribution of the zeros of the krawchkouk coefficients.
TL;DR: In this article, the indeterminate Hamburger moment problems associated with q-Laguerre polynomials are studied and the coefficients in their recurrence relations are of exponential growth.
TL;DR: In this paper, the expansion coefficients of xm and xmqj(x), m∈N0, in series of the set ∗p n (x)∗ are found in terms of the polynomials σ(x) and τ(x).
TL;DR: In this article, a one-parameter family of polynomials orthogonal on the unit circle (OPC) is constructed, where the value of the parameter defines the arc on the circle where the weight function vanishes.
01 Jan 1998
TL;DR: In this paper, the combinatorics of the Al-Salam-Chihara polynomials are given, and three classification theorems for generalized moments as orthogonal polynomial moments are proven.
Abstract: Classical orthogonal polynomials as moments for other classical orthogonal polynomials are obtained via linear functionals. The combinatorics of the Al-Salam-Chihara polynomials are given, and three classification theorems for generalized moments as orthogonal polynomials are proven. Some combinatorial explanations and open problems are discussed.
TL;DR: In this article, the authors obtained all spectral type differential equations satisfied by the Sobolev-type Laguerre polynomials, which generalizes the results found in 1990 by the first and second author in the case of the generalized Laguers.
Abstract: We obtain all spectral type differential equations satisfied by the Sobolev-type Laguerre polynomials. This generalizes the results found in 1990 by the first and second author in the case of the generalized Laguerre polynomials defined by T.H. Koornwinder in 1984.
TL;DR: In this article, it was shown that cubature formulae invariant under Z2 X • • • X Z2 are characterized by cubatures invariant on the unit sphere S in R. The results provide a new approach to study orthogonal polynomials and cubatures on spheres and on simplices.
Abstract: Orthogonal polynomials on the standard simplex E in R are shown to be related to the spherical orthogonal polynomials on the unit sphere S in R that are invariant under the group Z2 x • • • x Z2. For a large class of measures on S, cubature formulae invariant under Z2 X • • • X Z2 are shown to be characterized by cubature formulae on S. Moreover, it also is shown that there is a correspondence between orthogonal polynomials and cubature formulae on S and those invariant on the unit ball B in R. The results provide a new approach to study orthogonal polynomials and cubature formulae on spheres and on simplices.
TL;DR: In this article, it was shown that matrix harmonics on S2 (obtained from harmonic polynomials in 3 variables by replacing the commuting variables x1, x2, x3 by hermitian N×N matrices X1, X2, X3 satisfying ρ + cycl) are the constituents of a discrete Laplacian whose eigenvalues coincide with the first N 2 ones of the ordinary LaplACian on S 2.
Abstract: We show that matrix harmonics on S2 (obtained from harmonic polynomials in 3 variables by replacing the commuting variables x1, x2, x3 by hermitian N×N matrices X1, X2, X3 satisfying \(\), + cycl.) define two sets of families of discrete orthogonal polynomials, dual to each other, one of them having 3-term recurrence relations that, written in tridiagonal matrix form, are the constituents of a discrete Laplacian whose eigenvalues coincide with the first N2 ones of the ordinary Laplacian on S2.
TL;DR: In this article, a quadratically optimized polynomials for Monte Carlo simulations of quantum field theories with fermions are described and their implementation in the algebraic manipulation language Maple is discussed.
TL;DR: In this article, the authors studied the connection problem between discrete hypergeometric polynomials and the product case with m = 0 and its complete solution for all the classical discrete orthogonal hypergeometrical polynomial (CDOH) polynomorphisms is given.
TL;DR: In this article, a new family of polynomials which are eigenfunctions of a singular Sturm-Liouville problem on the triangle T 2 is described.
Abstract: In this paper we describe a new family of polynomials which are eigenfunctions of a singular Sturm–Liouville problem on the triangle T 2 = { ( x , y ) : x ≥ 0 , y ≥ 0 , x + y 1 } . The polynomials are shown to be orthogonal over T 2 with respect to a unit weight function, and may be used in approximations which are exponentially convergent for functions which are infinitely smooth in T 2 . The zeros of the polynomials may be used in cubature formulae on T 2 .