Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

AbstractA 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed thespectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to theq-Painlevé system.

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Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

Nagoya Mathematical Journal ◽

10.1215/00277630-3140952 ◽

2015 ◽

Vol 219 ◽

pp. 127-234 ◽

Cited By ~ 10

Author(s):

N. S. Witte

Keyword(s):

Orthogonal Polynomial ◽

Difference Operator ◽

Divided Difference ◽

Polynomial System ◽

Homogeneous Equation ◽

Polynomial Systems ◽

Classical Solutions ◽

First Order ◽

Wilson Operator ◽

Spectral Equation

AbstractA 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the q-Painlevé system.

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Stability Anomalies of Some Jacobian-Free Iterative Methods of High Order of Convergence

Axioms ◽

10.3390/axioms8020051 ◽

2019 ◽

Vol 8 (2) ◽

pp. 51

Author(s):

Alicia Cordero ◽

Javier G. Maimó ◽

Juan R. Torregrosa ◽

María P. Vassileva

Keyword(s):

Difference Operator ◽

Divided Difference ◽

Polynomial System ◽

Nonlinear Problems ◽

Coefficient Matrix ◽

Test Problems ◽

Stability Performance ◽

Need To Evaluate ◽

The Stability ◽

Theoretical Results

In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results.

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Characterization by orthogonal polynomial systems of finite Markov chains

Journal of Applied Probability ◽

10.1239/jap/1085496590 ◽

2001 ◽

Vol 38 (A) ◽

pp. 42-52 ◽

Cited By ~ 2

Author(s):

E. Seneta

Keyword(s):

Probability Distribution ◽

Markov Chains ◽

Orthogonal Polynomial ◽

Polynomial System ◽

Polynomial Systems ◽

Stochastic Matrices ◽

Mutation Model ◽

Hahn Polynomials ◽

Finite Markov Chains

The paper characterizes matrices which have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constraint a0j = 0 for j ≥ 2 are specified. The only stochastic matrices P = {pij} satisfying p00 + p01 = 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

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Characterization by orthogonal polynomial systems of finite Markov chains

Journal of Applied Probability ◽

10.1017/s0021900200112665 ◽

2001 ◽

Vol 38 (A) ◽

pp. 42-52 ◽

Cited By ~ 1

Author(s):

E. Seneta

Keyword(s):

Probability Distribution ◽

Markov Chains ◽

Orthogonal Polynomial ◽

Polynomial System ◽

Polynomial Systems ◽

Stochastic Matrices ◽

Mutation Model ◽

Hahn Polynomials ◽

Finite Markov Chains

The paper characterizes matriceswhich have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constrainta0j= 0 forj≥ 2 are specified. The only stochastic matricesP ={pij} satisfyingp00+p01= 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

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The Askey–Wilson polynomials and q-Sturm–Liouville problems

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100073916 ◽

1996 ◽

Vol 119 (1) ◽

pp. 1-16 ◽

Cited By ~ 17

Author(s):

B. Malcolm Brown ◽

W. Desmond Evans ◽

Mourad E. H. Ismail

Keyword(s):

Difference Operator ◽

Divided Difference ◽

Liouville Problem ◽

Inner Product ◽

Cauchy Principal ◽

Wilson Operator ◽

Principal Value ◽

Sturm Liouville ◽

Wilson Polynomials ◽

Theoretic Description

AbstractWe find the adjoint of the Askey–Wilson divided difference operator with respect to the inner product on L2(–1, 1, (1– x2)½dx) defined as a Cauchy principal value and show that the Askey-Wilson polynomials are solutions of a q-Sturm–Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the Askey-Wilson operator.

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Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421501236 ◽

2021 ◽

Vol 31 (09) ◽

pp. 2150123

Author(s):

Xiaoyan Chen ◽

Maoan Han

Keyword(s):

Limit Cycles ◽

Polynomial Systems ◽

Melnikov Function ◽

Unperturbed System ◽

Piecewise Polynomial ◽

First Order ◽

Number Of Zeros ◽

Period Annulus

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].

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Summation Kernels for Orthogonal Polynomial Systems

Handbook of Analytic-Computational Methods in Applied Mathematics ◽

10.1201/9780429123610-15 ◽

2019 ◽

pp. 709-749

Author(s):

Frank Filbir ◽

Rupert Lasser ◽

Josef Obermaier

Keyword(s):

Orthogonal Polynomial ◽

Polynomial Systems

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Bifurcation Analysis in a Class of Piecewise Nonlinear Systems with a Nonsmooth Heteroclinic Loop

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418500268 ◽

2018 ◽

Vol 28 (02) ◽

pp. 1850026

Author(s):

Yuanyuan Liu ◽

Feng Li ◽

Pei Dang

Keyword(s):

Limit Cycles ◽

Polynomial Systems ◽

Melnikov Function ◽

Heteroclinic Orbits ◽

Phase Portraits ◽

Unperturbed System ◽

Piecewise Polynomial ◽

Heteroclinic Loop ◽

First Order ◽

Nilpotent Critical Point

We consider the bifurcation in a class of piecewise polynomial systems with piecewise polynomial perturbations. The corresponding unperturbed system is supposed to possess an elementary or nilpotent critical point. First, we present 17 cases of possible phase portraits and conditions with at least one nonsmooth periodic orbit for the unperturbed system. Then we focus on the two specific cases with two heteroclinic orbits and investigate the number of limit cycles near the loop by means of the first-order Melnikov function, respectively. Finally, we take a quartic piecewise system with quintic piecewise polynomial perturbation as an example and obtain that there can exist ten limit cycles near the heteroclinic loop.

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