Recurrence Relation For Orthogonal Polynomials Research Articles

On the recurrence coefficients for the q-Laguerre weight and discrete Painlevé equations

Abstract We study the dependence of recurrence coefficients in the three-term recurrence relation for orthogonal polynomials with a certain deformation of the q-Laguerre weight on the degree parameter n. We show that this dependence is described by a discrete Painlevé equation on the family of A 5 ( 1 ) Sakai surfaces, but this equation is different from the standard examples of discrete Painlevé equations of this type and instead is a composition of two such. This case study is a good illustration of the effectiveness of a recently proposed geometric identification scheme for discrete Painlevé equations.

Two-dimensional contiguous relations for the linearization coefficients of classical orthogonal polynomials

ABSTRACT By using the three-term recurrence relation for orthogonal polynomials, we produce a collection of two-dimensional contiguous relations for certain generalized hypergeometric functions. These generalized hypergeometric functions arise through linearization coefficients for some classical orthogonal polynomials in the Askey-scheme, namely Gegenbauer (ultraspherical), Hermite, Jacobi and Laguerre polynomials.

A Generic and Strictly Banded Spectral Petrov–Galerkin Method for Differential Equations with Polynomial Coefficients

In this paper we describe a generic spectral Petrov–Galerkin method that is sparse and strictly banded for any linear ordinary differential equation with polynomial coefficients. The method applies to all subdivisions of Jacobi polynomials (e.g., Chebyshev and Legendre), utilizes well-known recurrence relations of orthogonal polynomials, and leads to almost exactly the same discretized system of equations as the integration preconditioners [E. A. Coutsias, T. Hagstrom, and D. Torres, Math. Comp., 65 (1996), pp. 611–635] if this method was redesigned to make use of trial functions that satisfy a given problem’s boundary conditions. A link between the new Petrov–Galerkin method and IP is revealed through a new recursion relation for Jacobi polynomials. Because of the strictly banded nature of all coefficient matrices, the new method extends easily and efficiently to multiple dimensions though the use of tensor product methods.

On the Extension of the Three-Term Recurrence Relation to Probabilities Distributions without Moments

In this paper, we extend the three-term recurrence relation for orthogonal polynomials associated with a probability distribution having a finite moment of all orders to a class of orthogonal functions associated with an infinitely divisible probability distribution µ having a finite moments of order less or equal to four. An explicit expression of these functions will be given in term of the Lévy-Khintchine function of the measure µ.

Recurrence relations for orthogonal polynomials for PDEs in polar and cylindrical geometries.

This paper introduces two families of orthogonal polynomials on the interval (−1,1), with weight function omega (x)equiv 1. The first family satisfies the boundary condition p(1)=0, and the second one satisfies the boundary conditions p(-1)=p(1)=0. These boundary conditions arise naturally from PDEs defined on a disk with Dirichlet boundary conditions and the requirement of regularity in Cartesian coordinates. The families of orthogonal polynomials are obtained by orthogonalizing short linear combinations of Legendre polynomials that satisfy the same boundary conditions. Then, the three-term recurrence relations are derived. Finally, it is shown that from these recurrence relations, one can efficiently compute the corresponding recurrences for generalized Jacobi polynomials that satisfy the same boundary conditions.

Recurrence Relations for Orthogonal Polynomials on Triangular Domains

In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n , r ( u , v , w ) , r = 0 , 1 , … , n , n ≥ 0 on the triangular domain T = { ( u , v , w ) : u , v , w ≥ 0 , u + v + w = 1 } are constructed, where u , v , w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is a need for a more efficient alternative. A very convenient method for computing orthogonal polynomials is based on recurrence relations. Such recurrence relations are described in this paper for the triangular orthogonal polynomials, providing a simple and fast algorithm for their evaluation.

Construction and implementation of asymptotic expansions for Jacobi–type orthogonal polynomials

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree $n$ goes to $\infty$. These are defined on the interval $[-1,1]$ with weight function $w(x)=(1-x)^{\alpha}(1+x)^{\beta}h(x)$, $\alpha,\beta>-1$ and $h(x)$ a real, analytic and strictly positive function on $[-1,1]$. This information is available in the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen, where the authors use the Riemann--Hilbert formulation and the Deift--Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in $\mathcal{O}(n)$ operations, rather than $\mathcal{O}(n^2)$ based on the recurrence relation.

Orthogonal Polynomials On Ellipses And Their Recurrence Relations

AbstractIn this note we study the connection between orthogonal polynomials on an ellipse and orthogonal Laurent polynomials on the unit circle relative to some multiplicative measures and then establish the recurrence relations for orthogonal polynomials on an ellipse. The matrix representation of the operator of multiplication by coordinate function is obtained

Algorithms for the Geronimus transformation for orthogonal polynomials on the unit circle

Let Lˆ be a positive definite bilinear functional on the unit circle defined on Pn, the space of polynomials of degree at most n. Then its Geronimus transformation L is defined by Lˆ(p,q)=L((z−α)p(z),(z−α)q(z)) for all p,q∈Pn, α∈C. Given Lˆ, there are infinitely many such L which can be described by a complex free parameter. The Hessenberg matrix that appears in the recurrence relations for orthogonal polynomials on the unit circle is unitary, and can be factorized using its associated Schur parameters. Recent results show that the unitary Hessenberg matrices associated with L and Lˆ, respectively, are related by a QR step where all the matrices involved are of order n+1. For the analogue on the real line of this so-called spectral transformation, the tridiagonal Jacobi matrices associated with the respective functionals are related by an LR step. In this paper we derive algorithms that compute the new Schur parameters after applying a Geronimus transformation. We present two forward algorithms and one backward algorithm. The QR step between unitary Hessenberg matrices plays a central role in the derivation of each of the algorithms, where the main idea is to do the inverse of a QR step. Making use of the special structure of unitary Hessenberg matrices, all the algorithms are efficient and need only O(n) flops. We present several numerical experiments to analyse the accuracy and to explain the behaviour of the algorithms.

Coherent orthogonal polynomials

We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we include thus–in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions–Lie algebra theory.We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis {|x〉}, for an alternative countable basis {|n〉}. The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively.Differential recurrence relations of orthogonal polynomials allow us to realize that they determine an infinite-dimensional irreducible representation of a non-compact Lie algebra, whose second order Casimir C gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl–Heisenberg algebra h(1) with C=0 for Hermite polynomials and su(1,1) with C=−1/4 for Laguerre and Legendre polynomials are obtained.Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the L2 functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space L2 and, in particular, generalized coherent polynomials are thus obtained.

The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation

We show that the coefficients of the three-term recurrence relation for orthogonal polynomials with respect to a semi-classical extension of the Laguerre weight satisfy the fourth Painlevé equation when viewed as functions of one of the parameters in the weight. We compare different approaches to derive this result, namely, the ladder operators approach, the isomonodromy deformations approach and combining the Toda system for the recurrence coefficients with a discrete equation. We also discuss a relation between the recurrence coefficients for the Freud weight and the semi-classical Laguerre weight and show how it arises from the Bäcklund transformation of the fourth Painlevé equation.

Solution of a cauchy problem for an infinite chain of linear differential equations

Defining the recurrence relations for orthogonal polynomials we have found an exact solution of a Cauchy problem for an infinite chain of linear differential equations with constant coefficients. These solutions have been found both for homogeneous and an inhomogeneous systems.

Orthogonal polynomials (in Matlab)

A suite of Matlab programs has been developed as part of the book “Orthogonal Polynomials: Computation and Approximation” Oxford University Press, Oxford, 2004, by Gautschi. The package contains routines for generating orthogonal polynomials as well as routines dealing with applications. In this paper, a brief review of the first part of the package is given, dealing with procedures for generating the three-term recurrence relation for orthogonal polynomials and more general recurrence relations for Sobolev orthogonal polynomials. Moment-based methods and discretization methods, and their implementation in Matlab, are among the principal topics discussed.

Symmetries of factorization chains for the discrete Schrödinger equation

Factorization chains for the one-dimensional discrete Schrödinger equation (or the three-term recurrence relation for orthogonal polynomials) defining discrete-time Toda and Volterra lattices are considered. Discrete symmetries, arising from a freedom in the intermediate steps of corresponding double spectral transformations, are described. Some consequences for orthogonal polynomials are discussed.

Companion orthogonal polynomials

We give some properties relating the recurrence relations of orthogonal polynomials associated with any two symmetric distributions d φ 1( x) and d 2( x) such that dφ 2( x) = (1 + kx 2)d 1( x). As applications of properties, recurrence relations for many interesting systems of orthogonal polynomials are obtained.

Vector Orthogonal Polynomials and Least Squares Approximation

We describe an algorithm for complex discrete least squares approximation, which turns out to be very efficient when function values are prescribed in points on the real axis or on the unit circle. In the case of polynomial approximation, this reduces to algorithms proposed by Rutishauser, Gragg, Harrod, Reichel, Ammar, and others. The underlying reason for efficiency is the existence of a recurrence relation for orthogonal polynomials, which are used to represent the solution. We show how these ideas can be generalized to least squares approximation problems of a more general nature.

Is the recurrence relation for orthogonal polynomials always stable?

Attention is drawn to a phenomenon of “pseudostability” in connection with the three-term recurrence relation for discrete orthogonal polynomials. The computational implications of this phenomenon are illustrated in the case of discrete Legendre and Krawtchouk polynomials. The phenomenon also helps to explain a form of instability in Stieltjes's procedure for generating recursion coefficients of discrete orthogonal polynomials.

Asymptotics for orthogonal polynomials on and off the essential spectrum

We assume that the coefficients in the three term recurrence relation for orthogonal polynomials form sequences of bounded variation and we show that the asymptotic behavior of the orthogonal polynomials on the essential spectrum is closely related to the asymptotic behavior in the complex plane (off the essential spectrum), proving a conjecture of Máté, Nevai and Totik [ Constr. Approx. 1 (1985) , 231–248].

Representation of Special Functions by Differintegral and Hyperdifferential Operators

The use of differintegral operators $D^\nu $ with $D = {d / {dz}}$ and $\nu \in \mathbb{C}$ and hyperdifferential operators, together with some operational calculus manipulations leads to new formulae of representation of usual special functions. These formulae are useful for obtaining generating functions and differential recurrence relations of orthogonal polynomials.

The triangular decomposition of Hankel matrices

An algorithm for determining the triangular decomposition H = R ∗ D R H = {R^ \ast }DR of a Hankel matrix H using O ( n 2 ) O({n^2}) operations is derived. The derivation is based on the Lanczos algorithm and the relation between orthogonalization of vectors and the triangular decomposition of moment matrices. The algorithm can be used to compute the three-term recurrence relation for orthogonal polynomials from a moment matrix.