801 research outputs found
Quadratic Hermite-Pade approximation to the exponential function: a Riemann-Hilbert approach
We investigate the asymptotic behavior of the polynomials p, q, r of degrees
n in type I Hermite-Pade approximation to the exponential function, defined by
p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are
characterized by a Riemann-Hilbert problem for a 3x3 matrix valued function. We
use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to
obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz),
and r(3nz) in every domain in the complex plane. An important role is played by
a three-sheeted Riemann surface and certain measures and functions derived from
it. Our work complements recent results of Herbert Stahl.Comment: 60 pages, 13 figure
Orthogonal polynomials with asymptotically periodic recurrence coefficients
AbstractGiven the coefficients in the three term recurrence relation satisfied by orthogonal polynomials, we investigate the properties of those classes of polynomials whose coefficients are asymptotically periodic. Assuming a rate of convergence of the coefficients to their asymptotic values, we construct the measure with respect to which the polynomials are orthogonal and discuss their asymptotic behavior
Asymptotic behaviour for Wall polynomials and the addition formula for little -Legendre polynomials
Asymptotics for Hermite-Pade rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)
We investigate the asymptotic behavior for type II Hermite-Pade approximation
to two functions, where each function has two branch points and the pairs of
branch points are separated. We give a classification of the cases such that
the limiting counting measures for the poles of the Hermite-Pade approximants
are described by an algebraic function of order 3 and genus 0. This situation
gives rise to a vector-potential equilibrium problem for three measures and the
poles of the common denominator are asymptotically distributed like one of
these measures. We also work out the strong asymptotics for the corresponding
Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that
characterizes this Hermite-Pade approximation problem.Comment: 102 pages, 31 figure
Some discrete multiple orthogonal polynomials
27 pages, no figures.-- MSC2000 codes: 33C45, 33C10, 42C05, 41A28.-- Issue title: "Proceedings of the 6th International Symposium on Orthogonal Polynomials, Special Functions and their Applications" (OPSFA-VI, Rome, Italy, 18-22 June 2001).MR#: MR1985676 (2004g:33015)Zbl#: Zbl 1021.33006In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn (T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978; R. Koekoek and R.F. Swarttouw, Reports of the Faculty of Technical Mathematics and Informatics No. 98-17, Delft, 1998; A.F. Nikiforov et al., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991). These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order difference equation, and an explicit expression from which the coefficients of the three-term recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied by Van Assche and Coussement (J. Comput. Appl. Math. 127 (2001) 317–347) and Aptekarev et al. (Multiple orthogonal polynomials for classical weights, manuscript). The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally, there also exists a recurrence relation of order r+1 for these multiple orthogonal polynomials of type II. We compute the coefficients of the recurrence relation explicitly when r=2.This research was supported by INTAS project 00-272, Dirección General de Investigación del Ministerio de Ciencia y Tecnología of Spain under grants BFM-2000-0029 and BFM-2000-0206-C04-01, Dirección General de Investigación de la Comunidad Autónoma de Madrid, and by project G.0184.02 of FWO-Vlaanderen.Publicad
Exactly Solvable Birth and Death Processes
Many examples of exactly solvable birth and death processes, a typical
stationary Markov chain, are presented together with the explicit expressions
of the transition probabilities. They are derived by similarity transforming
exactly solvable `matrix' quantum mechanics, which is recently proposed by
Odake and the author. The (-)Askey-scheme of hypergeometric orthogonal
polynomials of a discrete variable and their dual polynomials play a central
role. The most generic solvable birth/death rates are rational functions of
( being the population) corresponding to the -Racah polynomial.Comment: LaTeX, amsmath, amssymb, 24 pages, no figure
- …