scholarly journals An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 534
Author(s):  
Lino G. Garza ◽  
Luis E. Garza ◽  
Edmundo J. Huertas
Keyword(s):  
Difference Equation ◽  
Recurrence Relation ◽  
Positive Measure ◽  
Asymptotic Properties ◽  
Nonlinear Difference Equation ◽  
Sobolev Type ◽  
Real Numbers ◽  
Recurrence Coefficients ◽  
Algebraic Properties ◽  
Three Term Recurrence Relation

In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:p,qs=∫Rp(x)q(x)dμ(x)+M0p(0)q(0)+M1p′(0)q′(0), where p,q are polynomials, M0, M1 are non-negative real numbers and μ is a symmetric positive measure. These include a five-term recurrence relation, a three-term recurrence relation with rational coefficients, and an explicit expression for its norms. Moreover, we use these results to deduce asymptotic properties for the recurrence coefficients and a nonlinear difference equation that they satisfy, in the particular case when dμ(x)=e−x4dx.

scholarly journals Solution and Attractivity for a Rational Recursive Sequence

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
E. M. Elsayed
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Specific Form ◽  
Nonlinear Difference Equation ◽  
Real Numbers ◽  
Positive Real ◽  
Special Cases ◽  
Recursive Sequence

This paper is concerned with the behavior of solution of the nonlinear difference equation , where the initial conditions , , are arbitrary positive real numbers and are positive constants. Also, we give specific form of the solution of four special cases of this equation.

scholarly journals Dynamics and Solutions of a Fifth-Order Nonlinear Difference Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-21 ◽  
Author(s):  
M. M. El-Dessoky ◽  
E. M. Elabbasy ◽  
Asim Asiri
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Nonlinear Difference Equation ◽  
Real Numbers ◽  
Positive Real ◽  
Special Cases ◽  
Rational Difference Equation ◽  
Fifth Order

The main objective of this paper is to study the behavior of the rational difference equation of the fifth-order yn+1=αyn+βynyn-3/(Ayn-4+Byn-3), n=0,1,…, where α,β,A, and B are real numbers and the initial conditions y-4,y-3,y-2,y-1 and y0 are positive real numbers such that Ay-4+By-3≠0. Also, we obtain the solution of some special cases of this equation.

On Orthogonal Polynomials With Respect to a Positive Definite Matrix of Measures

1995 ◽  
Vol 47 (1) ◽  
pp. 88-112 ◽  
Author(s):  
Antonio J. Duran
Keyword(s):  
Orthogonal Polynomials ◽  
Recurrence Relation ◽  
Positive Measure ◽  
Asymptotic Properties ◽  
Positive Definite Matrix ◽  
Positive Definite

AbstractIn this paper, we prove that any sequence of polynomials (pn)n for which dgr(pn) = n which satisfies a (2N + l)-term recurrence relation is orthogonal with respect to a positive definite N × N matrix of measures. We use that result to prove asymptotic properties of the kernel polynomials associated to a positive measure or a positive definite matrix of measures. Finally, some examples are given.

scholarly journals Global Stability of a Rational Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Guo-Mei Tang ◽  
Lin-Xia Hu ◽  
Gang Ma
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Stability ◽  
Positive Solutions ◽  
Open Problem ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Nonlinear Difference Equation ◽  
Real Numbers ◽  
Rational Difference Equation

We consider the higher-order nonlinear difference equation with the parameters, and the initial conditions are nonnegative real numbers. We investigate the periodic character, invariant intervals, and the global asymptotic stability of all positive solutions of the above-mentioned equation. In particular, our results solve the open problem introduced by Kulenović and Ladas in their monograph (see Kulenović and Ladas, 2002).

scholarly journals Ratio asymptotics for biorthogonal matrix polynomials with unbounded recurrence coefficients

2020 ◽  
pp. 51-51
Author(s):  
Amílcar Branquinho ◽  
Juan Garca-Ardila ◽  
Francisco Marcellán
Keyword(s):  
Recurrence Relation ◽  
Recurrence Coefficients ◽  
Matrix Polynomials ◽  
Ratio Asymptotics ◽  
Matrix Coefficients ◽  
Biorthogonal Polynomials ◽  
Matrix Biorthogonal Polynomials ◽  
Three Term Recurrence Relation

In this paper we study matrix biorthogonal polynomials sequences that satisfy a nonsymmetric three term recurrence relation with unbounded matrix coefficients. The outer ratio asymptotics for this family of matrix biorthogonal polynomials is derived under quite general assumptions. Some illustrative examples are considered.

scholarly journals Dynamics of a Higher-Order Nonlinear Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Guo-Mei Tang ◽  
Lin-Xia Hu ◽  
Xiu-Mei Jia
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Initial Conditions ◽  
Higher Order ◽  
Nonlinear Difference Equation ◽  
Asymptotically Stable ◽  
Unique Equilibrium ◽  
Real Numbers ◽  
Positive Real ◽  
Globally Asymptotically Stable

We consider the higher-order nonlinear difference equation , , where parameters are positive real numbers and initial conditions are nonnegative real numbers, . We investigate the periodic character, the invariant intervals, and the global asymptotic stability of all positive solutions of the abovementioned equation. We show that the unique equilibrium of the equation is globally asymptotically stable under certain conditions.

scholarly journals An ubiquitous three-term recurrence relation

2021 ◽  
Vol 62 (3) ◽  
pp. 032106
Author(s):  
Paolo Amore ◽  
Francisco M. Fernández
Keyword(s):  
Recurrence Relation ◽  
Three Term Recurrence Relation

DRY TURBULENCE FROM A TIME-DELAYED CHUA'S CIRCUIT

1993 ◽  
Vol 03 (02) ◽  
pp. 645-668 ◽  
Author(s):  
A. N. SHARKOVSKY ◽  
YU. MAISTRENKO ◽  
PH. DEREGEL ◽  
L. O. CHUA
Keyword(s):  
Difference Equation ◽  
Stability Region ◽  
Nonlinear Difference Equation ◽  
Chua’S Circuit ◽  
Stability Boundaries ◽  
Chua's Circuit ◽  
Infinite Dimensional ◽  
Chaotic Region ◽  
The Stability ◽  
Left Portion

In this paper, we consider an infinite-dimensional extension of Chua's circuit (Fig. 1) obtained by replacing the left portion of the circuit composed of the capacitance C2 and the inductance L by a lossless transmission line as shown in Fig. 2. As we shall see, if the remaining capacitance C1 is equal to zero, the dynamics of this so-called time-delayed Chua's circuit can be reduced to that of a scalar nonlinear difference equation. After deriving the corresponding 1-D map, it will be possible to determine without any approximation the analytical equation of the stability boundaries of cycles of every period n. Since the stability region is nonempty for each n, this proves rigorously that the time-delayed Chua's circuit exhibits the "period-adding" phenomenon where every two consecutive cycles are separated by a chaotic region.

Orthogonal polynomials: applications and computation

Acta Numerica ◽  
1996 ◽  
Vol 5 ◽  
pp. 45-119 ◽  
Author(s):  
Walter Gautschi
Keyword(s):  
Numerical Methods ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Rational Function ◽  
Recurrence Relations ◽  
Recurrence Coefficients ◽  
Basic Task ◽  
Sobolev Orthogonal Polynomials ◽  
Gauss Type ◽  
Three Term Recurrence Relation

We give examples of problem areas in interpolation, approximation, and quadrature, that call for orthogonal polynomials not of the classical kind. We then discuss numerical methods of computing the respective Gauss-type quadrature rules and orthogonal polynomials. The basic task is to compute the coefficients in the three-term recurrence relation for the orthogonal polynomials. This can be done by methods relying either on moment information or on discretization procedures. The effect on the recurrence coefficients of multiplying the weight function by a rational function is also discussed. Similar methods are applicable to computing Sobolev orthogonal polynomials, although their recurrence relations are more complicated. The paper concludes with a brief account of available software.

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