Term Recurrence Relation Research Articles

Zeros of polynomials embedded in an orthogonal sequence

Let $\{x_{k,n}\}_{k=1}^n$ and $\{x_{k,n+1}\}_{k=1}^{n+1}$ , n????, be two given sets of real distinct points with x 1,n?+?1? ?0, which are chosen arbitrarily, that define p n?+?2 via the three term recurrence relation $$ p_{n+2}(x)=(x-b_{n+1})p_{n+1}(x)-\lambda_{n+1}p_n(x). $$

An accurate Fortran code for computing hydrogenic continuum wave functions at a wide range of parameters

Abstract The accurate computations of hydrogenic continuum wave functions are very important in many branches of physics such as electron-atom collisions, cold atom physics, and atomic ionization in strong laser fields, etc. Although there already exist various algorithms and codes, most of them are only reliable in a certain ranges of parameters. In some practical applications, accurate continuum wave functions need to be calculated at extremely low energies, large radial distances and/or large angular momentum number. Here we provide such a code, which can generate accurate hydrogenic continuum wave functions and corresponding Coulomb phase shifts at a wide range of parameters. Without any essential restrict to angular momentum number, the present code is able to give reliable results at the electron energy range [ 10 − 3 , 10 3 ] eV for radial distances of [ 10 − 2 , 10 4 ] a.u. We also find the present code is very efficient, which should find numerous applications in many fields such as strong field physics. Program summary Program title: HContinuumGautchi Catalogue identifier: AEHD_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHD_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 1233 No. of bytes in distributed program, including test data, etc.: 7405 Distribution format: tar.gz Programming language: Fortran90 in fixed format Computer: AMD Processors Operating system: Linux RAM: 20 MBytes Classification: 2.7, 4.5 Nature of problem: The accurate computation of atomic continuum wave functions is very important in many research fields such as strong field physics and cold atom physics. Although there have already existed various algorithms and codes, most of them can only be applicable and reliable in a certain range of parameters. We present here an accurate FORTRAN program for calculating the hydrogenic continuum wave functions in a very wide range of parameters, which suffices the needs of most practical applications. The Coulomb phases are also calculated. For any given momentum, radial point, and the largest angular momentum number, the code calculates all the angular components at once. The algorithm we adopt has been given in details by Gautchi [1,2], who suggested a stable minimal solution of general three term recurrence relations. Solution method: Minimum solution of three-term recurrence relations developed by W. Gautchi [1,2]. Running time: A few seconds to a few minutes, depending how many different wave functions one needs to calculate.

A canonical family of multiple orthogonal polynomials for Nikishin systems

For any pair of compact intervals of the real line Δ 1 , Δ 2 , with Δ 1 ∩ Δ 2 = ∅ , we obtain two probability measures μ 1 , τ 1 , supported on Δ 1 and Δ 2 respectively, such that the Nikishin system N ( μ 1 , τ 1 ) has a sequence of monic multiple orthogonal polynomials which satisfy a four term recurrence relation with constant coefficients of period 2. The measures are obtained from the functions which give the ratio asymptotic of multiple orthogonal polynomials with respect to an arbitrary Nikishin system N ( σ 1 , σ 2 ) on Δ 1 , Δ 2 , such that σ i ′ > 0 a.e. on Δ i , i = 1 , 2 . The role of μ 1 , τ 1 is symmetric in the sense that the same construction is possible on Δ 2 , Δ 1 , with N ( τ 1 , μ 1 ) .

A Characterization of L-orthogonal Polynomials from Three Term Recurrence Relations

We consider the sequence of polynomials {Q n } satisfying the L-orthogonality ?[z ?n+m Q n (z)]=0, 0?m?n?1, with respect to a linear functional ? for which the moments ?[t n ]=μ n are all complex. Under certain restriction on the moment functional these polynomials also satisfy a three term recurrence relation. We consider three special classes of such moment functionals and characterize them in terms of the coefficients of the associated three term recurrence relations. Relations between the polynomials {Q n } associated with two of these special classes of moment functionals are also given. Examples are provided to justify this characterization.

New orthogonality relations for the continuous and the discrete q-ultraspherical polynomials

In a recent contribution [N.M. Atakishiyev, A.U. Klimyk, On discrete q-ultraspherical polynomials and their duals, J. Math. Anal. Appl. 306 (2005) 637–645], the so-named discrete q-ultraspherical polynomials were introduced as a specialization of the big q-Jacobi polynomials, and their orthogonality established for values of the parameter outside its commonly known domain but inside the range of validity of the conditions of Favard's theorem. In this paper we consider both the continuous and the discrete q-ultraspherical polynomials and we prove that their orthogonality is guaranteed for the whole range of the allowed parameters, even in those intriguing cases in which the three term recurrence relation breaks down. The presence of either the Askey–Wilson divided difference operator (in the continuous case), or the q-derivative operator (in the discrete one), provides the q-Sobolev character of the non-standard inner products introduced in our approach.

Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s (d + 1 ) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials. We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions. We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite-Padé approximation in matrix form is given.

Three term recurrence for the evaluation of multivariate orthogonal polynomials

In this paper we obtain some explicit three term recurrence relations for the determination of multivariate orthogonal polynomials. These formulas allow us to obtain evaluation algorithms of finite series of these polynomials.

Matrix polynomials satisfying first order differential equations and three term recurrence relations

We describe families of matrix valued polynomials satisfying simultaneously a first order differential equation and a three term recurrence relation. Our goal is to address the classification of the matrix valued polynomials satisfying first order differential equations through the solutions of the so-called bispectral problem. At the heart of this lies the need to solve some complicated nonlinear equations with matrix coefficients called ad-conditions. The solutions of these equations are studied under a variety of sufficient conditions on its coefficients.

A fast algorithm for evaluation of normalized Hermite functions

An algorithm for computing the normalized Hermite Functions, h n (x) in floating point arithmetic is presented. The algorithm is based on an efficient numerical evaluation of certain closed contour integrals in the complex plane. For large degree n, the algorithm is significantly faster than the O(n) complexity of the well known three term recurrence relation. Comparable accuracy is achieved in no more than $O(\sqrt{n})$ operations, and for arguments bounded away from $\pm \sqrt {2n}$ , only $O(\sqrt{\ln{n}})$ operations.

A class of matrix-valued polynomials generalizing Jacobi polynomials

A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a three term recurrence relation, integral inter-relations, and weak orthogonality relations.

Variable step size initial value algorithm for singular perturbation problems using locally exact integration

We consider quasilinear singular perturbation problems of the form ε y ″ + p ( x ) y ′ + q ( x , y ) = h ( x ) , x ∈ [ 0 , 1 ] ; y ( 0 ) = α , y ( 1 ) = β with a boundary layer at one end point. The original problem is reduced to an asymptotically equivalent linear first order initial-value problem (IVP). Then, a variable step size initial value algorithm is applied to solve this (IVP). The algorithm is based on the locally exact integration of quadratic linearized problem coefficients on a non-uniform mesh. Two term-recurrence relation with controlled step size is obtained. Several problems are solved to demonstrate the applicability and efficiency of the algorithm. It is observed that the present method approximates the exact solution very well.

QES SYSTEMS, INVARIANT SPACES AND POLYNOMIALS RECURSIONS

Let us denote [Formula: see text], the finite-dimensional vector spaces of functions of the form ψ(x) = pn(x)+f(x)pm(x), where pn(x) and pm(x) are arbitrary polynomials of degree at most n and m in the variable x while f(x) represents a fixed function of x. Conditions on m, n and f(x) are found such that families of linear differential operators exist which preserve [Formula: see text]. A special emphasis is accorded to the cases where the set of differential operators represents the enveloping algebra of some abstract algebra. These operators can be transformed into linear matrix valued differential operators. In the second part, such types of operators are considered and a connection is established between their solutions and series of polynomials-valued vectors obeying three terms recurrence relations. When the operator is quasi-exactly solvable, it possesses a finite-dimensional invariant vector space. We study how this property leads to the truncation of the polynomials series.

Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions

Three term recurrence relations $y_{n+1}+b_n y_n+a_n y_{n-1}=0$ can be used for computing recursively a great number of special functions. Depending on the asymptotic nature of the function to be computed, different recursion directions need to be considered: backward for minimal solutions and forward for dominant solutions. However, some solutions interchange their role for finite values of $n$ with respect to their asymptotic behaviour and certain dominant solutions may transitorily behave as minimal. This phenomenon, related to Gautschi’s anomalous convergence of the continued fraction for ratios of confluent hypergeometric functions, is shown to be a general situation which takes place for recurrences with $a_n$ negative and $b_n$ changing sign once. We analyze the anomalous convergence of the associated continued fractions for a number of different recurrence relations (modified Bessel functions, confluent and Gauss hypergeometric functions) and discuss the implication of such transitory behaviour on the numerical stability of recursion.

Relations between univalent and orthogonal polynomials

Two sequences of polynomials are studied. One satisfies a three term recurrence relation for specific parameters and another a para-orthogonality property. Using the fact that these polynomials have their zeros lying on the unit circle and some other properties, we establish a criterion in order that the polynomials be univalent in the open unit disk.

A seventh order numerical method for singular perturbation problems

In this paper, a seventh order numerical method is presented for solving singularly perturbed two-point boundary value problems with a boundary layer at one end point. The two-point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a seventh order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two-point boundary value problem is obtained from the theory of singular perturbations. It is used in the seventh order compact difference scheme to get a two term recurrence relation and is solved. Several linear and nonlinear singular perturbation problems have been solved and the numerical results are presented to support the theory.

A stable recurrence for the incomplete gamma function with imaginary second argument

Even though the two term recurrence relation satisfied by the incomplete gamma function is asymptotically stable in at least one direction, for an imaginary second argument there can be a considerable loss of correct digits before stability sets in. We present an approach to compute the recurrence relation to full precision, also for small values of the arguments, when the first argument is negative and the second one is purely imaginary. A detailed analysis shows that this approach works well for all values considered.

Asymptotic behavior of solutions of general three term recurrence relations

We consider the solutions of general three term recurrence relations whose coefficients are analytic functions in a prescribed region. We study the ratio asymptotic of such solutions under the assumption that the coefficients are asymptotically periodic, and their strong asymptotic under more restrictive conditions.

Structure of certain Chebyshev-type polynomials in Onsager's algebra representation

In this report, we present a systematic account of mathematical structures of certain special polynomials arisen from the energy study of the superintegrable N-state chiral Potts model with a finite number of sizes. The polynomials of low-lying sectors are represented in two different forms, one of which is directly related to the energy description of superintegrable chiral Potts Z N -spin chain via the representation theory of Onsager's algebra. Both two types of polynomials satisfy some ( N + 1 ) -term recurrence relations, and Nth-order differential equations; polynomials of one kind reveal certain Chebyshev-like properties. Here, we provide a rigorous mathematical argument for cases N = 2 , 3 , and further raise some mathematical conjectures on those special polynomials for a general N.

A basic class of symmetric orthogonal polynomials using the extended Sturm–Liouville theorem for symmetric functions

In this research, by applying the extended Sturm–Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation along with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on, are presented. Then, it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class. They are respectively the generalized ultraspherical polynomials, generalized Hermite polynomials and two other sequences of symmetric polynomials, which are finitely orthogonal on ( − ∞ , ∞ ) and can be expressed in terms of the mentioned class directly. In this way, two half-trigonometric sequences of orthogonal polynomials, as special sub-cases of BCSOP, are also introduced.

Gauss quadrature rules for a generalized Hermite weight function

In this paper we give an algorithm for finding the coefficients in the three term recurrence relation for the polynomials orthogonal with respect to the weight function W( x) = ∣ x − b∣ 2 α exp[−( x + c) 2] on R , where α , b , c ∈ R , 2 α > −1. The algorithm is based on the discretization technique proposed by Gautschi, with the Lanczos algorithm. Using the coefficients obtained by this algorithm we construct the Gauss quadrature rules relative to the weight function W( x). To show the performance of the new algorithm we give some numerical examples.