Showing papers on "Discrete orthogonal polynomials published in 2008"
30 Jan 2008
TL;DR: A survey of the analytic theory of matrix orthogonal polynomials is given in this article, with a focus on block Jacobi matrices and block CMV matrices.
Abstract: We survey the analytic theory of matrix orthogonal polynomials. MSC: 42C05, 47B36, 30C10 keywords: orthogonal polynomials, matrix-valued measures, block Jacobi matrices, block CMV matrices
201 citations
TL;DR: In this paper, a systemic study of some families of multiple q-Bernoulli numbers and polynomials by using the multivariate q-Volkenborn integral is presented.
Abstract: A purpose of this paper is to present a systemic study of some families of multiple q-Bernoulli numbers and polynomials by using the multivariate q-Volkenborn integral (= p-adic q-integral) on ℤ
p
. Moreover, the study of these higher-order q-Bernoulli numbers and polynomials implies some interesting q-analogs of Stirling number identities.
157 citations
TL;DR: In this paper, a strong linearization of regular matrix polynomials P(λ) when represented in various polynomial bases (other than the monomials 1, λ, etc) is presented.
Abstract: This paper concerns regular matrix polynomials P(λ) when represented in various polynomial bases (other than the monomials 1, λ, λ 2 ,...). As in the monomial case, matrices of 'companion' form play an important part in theory and numerical practice. In particular, they are used here to construct 'strong linearizations' of P(λ). The paper contains three theorems concerning linearizations constructed for representations in a general class of 'degree-graded' polynomials, Bernstein polynomials and Lagrange polynomials.
133 citations
TL;DR: Uniform approximation of differentiable or analytic functions of one or several variables on a compact set K is studied by a sequence of discrete least squares polynomials if K satisfies a Markov inequality and point evaluations on standard discretization grids provide nearly optimal approximants.
120 citations
TL;DR: In this paper, the authors deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erd˝ os and Turan.
Abstract: In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erd˝ os and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomial.
103 citations
TL;DR: A nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) to K-theoretic Grassmannian Littlewood-Richardson rule of Buch and Buch (Acta Math 189(1):37-78, 2002) was proposed in this paper.
Abstract: We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) in the basis of stable Grothendieck polynomials for partitions This gives a common generalization, as well as new proofs of the rule of Fomin and Greene (Discret Math 193:565–596, 1998) for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood–Richardson rule of Buch (Acta Math 189(1):37–78, 2002) The proof is based on a generalization of the Robinson–Schensted and Edelman–Greene insertion algorithms Our results are applied to prove a number of new formulas and properties for K-theoretic quiver polynomials, and the Grothendieck polynomials of Lascoux and Schutzenberger (C R Acad Sci Paris Ser I Math 294(13):447–450, 1982) In particular, we provide the first K-theoretic analogue of the factor sequence formula of Buch and Fulton (Invent Math 135(3):665–687, 1999) for the cohomological quiver polynomials
90 citations
TL;DR: In this article, a theory of multivariate stable polynomials was developed for solving Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables.
Abstract: In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Polya-Schur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables. In particular, we answer a question of Hinkkanen on multivariate apolarity.
89 citations
TL;DR: A new biorthogonal matrix ensemble is introduced, namely the chiral unitary perturbed by a source term, whose multiple polynomials are related to the modified Bessel function of the first kind.
87 citations
TL;DR: In this article, a class of generalized complex polynomials of Hermite type, suggested by a special magnetic Schrodinger operator, is introduced and some related basic properties are discussed.
82 citations
TL;DR: The Bell polynomials and the binomial type sequences are studied to establish some relations tied to these important concepts and to deduce some interesting relations which enable us to obtain some new identities for the Bell poynomials.
67 citations
TL;DR: The main purpose of this paper is to prove an identity of symmetry for the higher order Bernoulli polynomials.
TL;DR: In this article, the Voronovskaya type theorem and saturation of convergence for q-Bernstein polynomials for a function analytic in the disc U R : = { z : | z | R } (R > q ) for arbitrary fixed q ⩾ 1.
TL;DR: Two classes of (hyper)graph polynomials definable in second order logic are introduced, and a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities are outlined.
Abstract: We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities.
TL;DR: In this paper, a probabilistic approach to the generation of polyno-mials in two discrete variables is presented, which leads to a new class of orthogonal polynomials.
Abstract: We present here a probabilistic approach to the generation of new polyno- mials in two discrete variables. This extends our earlier work on the 'classical' orthogonal polynomials in a previously unexplored direction, resulting in the discovery of an exactly soluble eigenvalue problem corresponding to a bivariate Markov chain with a transition kernel formed by a convolution of simple binomial and trinomial distributions. The solution of the relevant eigenfunction problem, giving the spectral resolution of the kernel, leads to what we believe to be a new class of orthogonal polynomials in two discrete variables. Possibilities for the extension of this approach are discussed.
TL;DR: A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed in this paper, which share many algebraic and number theoretical properties with the classical Bernoullians.
TL;DR: The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed in this article, where vortex configurations with vortices of the same strength but positive or negative orientation are given by the zeros of the Adler-Moser polynomial, which arise in the description of rational solutions of the Korteweg-de Vries equation.
Abstract: The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are given by the zeros of the Adler-Moser polynomials, which arise in the description of rational solutions of the Korteweg-de Vries equation. For quadupole background flow, vortex configurations are given by the zeros of polynomials expressed as wronskians of Hermite polynomials. Further new solutions are found in this case using the special polynomials arising the in the description of rational solutions of the fourth Painleve equation.
TL;DR: In this paper, the authors established the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior.
Abstract: We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples of multiple orthogonal polynomials, in particular Jacobi-Pineiro, Laguerre I and the example associated with modified Bessel functions. We also discuss an application to Toeplitz matrices.
TL;DR: It can be found that many results obtained before are special cases of these two relationships between the generalized Apostol-Bernoulli and apostol-Euler polynomials.
Abstract: The main object of this paper is to investigate the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials. We first establish two relationships between the generalized Apostol-Bernoulli and Apostol-Euler polynomials. It can be found that many results obtained before are special cases of these two relationships. Moreover, we have a study on the sums of products of the Apostol-Bernoulli polynomials and of the Apostol-Euler polynomials.
TL;DR: In this paper, the problem of finding closed formulas for the connection coefficients between orthogonal polynomials and the canonical sequence was studied, using a recurrence relation fulfilled by these coefficients and symbolic computation with the Mathematica language.
Abstract: We deal with the problem of obtaining closed formulas for the connection coefficients between orthogonal polynomials and the canonical sequence. We use a recurrence relation fulfilled by these coefficients and symbolic computation with the Mathematica language. We treat the cases of Gegenbauer, Jacobi and a new semi-classical sequence.
TL;DR: In this article, two families of rational solutions and associated special polynomials for the equations in the symmetric fourth Painleve hierarchy are studied, and the structure of the roots of these polynomial is shown to be highly regular in the complex plane.
Abstract: In this paper two families of rational solutions and associated special polynomials for the equations in the symmetric fourth Painleve hierarchy are studied. The structure of the roots of these polynomials is shown to be highly regular in the complex plane. Further representations are given of the associated special polynomials in terms of Schur functions. The properties of these polynomials are compared and contrasted with the special polynomials associated with rational solutions of the fourth Painleve equation.
TL;DR: In this paper, the authors simplify the definition of classical orthogonal polynomials given by Atakishiyev et al., 1995, and prove that the product of two functions is a solution of a fourth-order linear homogeneous divided-difference equation.
Abstract: By the study of various properties of some divided-difference equations, we simplify the definition of classical orthogonal polynomials given by Atakishiyev et al., 1995, On classical orthogonal polynomials, Constructive Approximation, 11, 181–226, then prove that orthogonal polynomials obtained by some modifications of the classical orthogonal polynomials on nonuniform lattices satisfy a single fourth-order linear homogeneous divided-difference equation with polynomial coefficients. Moreover, we factorize and solve explicitly these divided-difference equations. Also, we prove that the product of two functions, each of which satisfying a second-order linear homogeneous divided-difference equation is a solution of a fourth-order linear homogeneous divided-difference equation. This result holds in particular when the divided-difference operator is carefully replaced by the Askey–Wilson operator , following pioneering work by Magnus 1988, Associated Askey–Wilson polynomials as Laguerre–Hahn orthogonal polyno...
TL;DR: Examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order are presented.
TL;DR: Results for the interlacing of zeros of Jacobi polynomials of the same or adjacent degree are proved as one or both of the parameters are shifted continuously within a certain range.
Abstract: We prove results for the interlacing of zeros of Jacobi polynomials of the same or adjacent degree as one or both of the parameters are shifted continuously within a certain range. Numerical examples are given to illustrate situations where interlacing fails to occur.
TL;DR: The Bjorck-Pereyra algorithm, the Traub algorithm, certain new digital filter structures, as well as QR and divide and conquer eigenvalue algorithms are discussed, to obtain true generalizations of several algorithms.
TL;DR: In this article, the authors use orthogonality measures of Askey-Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances.
Abstract: We use orthogonality measures of Askey--Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey--Wilson polynomials are orthogonal martingale polynomials for these processes.
16 Dec 2008
TL;DR: A fundamentally new algebraic approach to the analysis and synthesis of discrete orthogonal basis functions and the concept of anisotropic moments is introduced and applied to 2D seismic data, which is an image processing problem.
Abstract: This paper presents a fundamentally new algebraic approach to the analysis and synthesis of discrete orthogonal basis functions. It provides the theoretical background to unify Fourier, Gabor and discrete orthogonal polynomial moments. For the first time, a set of objective tests are proposed to measure the quality of basis functions. It consists of two main sections: the theoretical background on the generation and orthogonalization of basis functions together with a new solution for the computation of spectra from incomplete data, as well as the implementation of interpolation for all orthogonal basis functions; a new approach to discrete orthogonal polynomials, proving that there is one and only one unitary discrete polynomial basis. Furthermore, the concept of anisotropic moments is introduced and applied to 2D seismic data, which is an image processing problem. The new polynomial basis is numerically better conditioned than the discrete cosine transform. This opens the door to new image compression algorithms, offering a higher compression ratio than the well known JPEG method, for the same numerical effort.
TL;DR: The q difference analog of the classical ladder operators was derived for those orthogonal polynomials arising from a class of indeterminate moments problems in this paper, where the ladder operators were used to solve the problem.
TL;DR: In this article, a Favard-type recursion relation for orthogonal free Sheffer polynomials with respect to a state is shown. But the relation is restricted to the case where the state is a rotation of a free product state.
Abstract: The subject of this paper are polynomials in multiple non-commuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favard-type recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in non-commuting variables with a resolvent-type generating function. Among such families, we describe the ones that are orthogonal. Their recursion relations have a more special form; the best way to describe them is in terms of the free cumulant generating function of the state of orthogonality, which turns out to satisfy a type of second-order difference equation. If the difference equation is in fact first order, and the state is tracial, we show that the state is necessarily a rotation of a free product state. We also describe interesting examples of non-tracial infinitely divisible states with orthogonal free Sheffer polynomials.
TL;DR: A survey on Chebyshev polynomials of third and fourth kind, which are respectively orthogonal with respect to the weight functions ρ 1 and ρ 2, to satisfy a semi minimax property that has application in approximating the functions of type Q ( x) P n ( x ) where P n is an arbitrary polynomial of degree n and Q (x ) denotes a constant weighting factor.
TL;DR: In this paper, the Estrada index of a graph whose eigenvalues are λ 1, λ 2, λ 3, ν n, ν 4, ξ 5, and ν 6 is approximated to ∑ i = 1 n e λ i.