scholarly journals DIFFERENCE EQUATION OF THE COLORED JONES POLYNOMIAL FOR TORUS KNOT

2004 ◽  
Vol 15 (09) ◽  
pp. 959-965 ◽  
Author(s):  
KAZUHIRO HIKAMI
Keyword(s):  
Difference Equation ◽  
Jones Polynomial ◽  
Order Difference ◽  
Colored Jones Polynomial ◽  
First Order ◽  
Torus Knot ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Hypergeometric Type ◽  
The Difference

We prove that the N-colored Jones polynomial for the torus knot [Formula: see text] satisfies the second order difference equation, which reduces to the first order difference equation for a case of [Formula: see text]. We show that the A-polynomial of the torus knot can be derived from the difference equation. Also constructed is a q-hypergeometric type expression of the colored Jones polynomial for [Formula: see text].

scholarly journals Global Dynamics of Delayed Sigmoid Beverton–Holt Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Toufik Khyat ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Global Dynamics ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Difference

In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation xn+1=fxn,xn−1, n=0,1,…, where f is decreasing in the variable xn and increasing in the variable xn−1. As a case study, we use the difference equation xn+1=xn−12/cxn−12+dxn+f, n=0,1,…, where the initial conditions x−1,x0≥0 and the parameters satisfy c,d,f>0. In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations.

scholarly journals Growth of Meromorphic Solutions of Some -Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Guowei Zhang
Keyword(s):  
Difference Equation ◽  
Fixed Points ◽  
Difference Equations ◽  
Meromorphic Solutions ◽  
Order Difference ◽  
Complex Difference ◽  
Complex Difference Equations ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Difference

We estimate the growth of the meromorphic solutions of some complex -difference equations and investigate the convergence exponents of fixed points and zeros of the transcendental solutions of the second order -difference equation. We also obtain a theorem about the -difference equation mixing with difference.

scholarly journals Kolmogorov-Arnold-Moser Theory and Symmetries for a Polynomial Quadratic Second Order Difference Equation

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 790
Author(s):  
Tarek F. Ibrahim ◽  
Zehra Nurkanović
Keyword(s):  
Difference Equation ◽  
Equilibrium Points ◽  
Kam Theory ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Elliptic Equilibrium ◽  
Order Difference Equation ◽  
The Difference ◽  
The Stability ◽  
Theoretical Results

By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of two elliptic equilibrium points (zero equilibrium and negative equilibrium) of the difference equation t n + 1 = α t n + β t n 2 − t n − 1 , n = 0 , 1 , 2 , … , where are t − 1 , t 0 , α ∈ R , α ≠ 0 , β > 0 . By using the symmetries we find the periodic solutions with some periods. Finally, some numerical examples are given to verify our theoretical results.

scholarly journals On the Behaviour of the Solutions of a Second-Order Difference Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
Leonid Gutnik ◽  
Stevo Stevic
Keyword(s):  
Difference Equation ◽  
Explicit Formula ◽  
Second Order ◽  
Order Difference ◽  
Monotone Solution ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Difference

We study the difference equationxn+1=α−xn/xn−1,n∈ℕ0, whereα∈ℝand wherex−1andx0are so chosen that the corresponding solution(xn)of the equation is defined for everyn∈ℕ. We prove that whenα=3the equilibriumx¯=2of the equation is not stable, which corrects a result due to X. X. Yan, W. T. Li, and Z. Zhao. For the caseα=1, we show that there is a strictly monotone solution of the equation, and we also find its asymptotics. An explicit formula for the solutions of the equation are given for the caseα=0.

scholarly journals TRANSFORMATION OF THE LINEAR DIFFERENCE EQUATION INTO A SYSTEM OF THE FIRST ORDER DIFFERENCE EQUATIONS

2019 ◽  
pp. 76-80
Author(s):  
M.I. Ayzatsky
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Linear Difference Equation ◽  
Second Order ◽  
Dimensional System ◽  
Order Difference ◽  
Order Linear ◽  
First Order ◽  
Second Order Difference Equation ◽  
Order Difference Equation

The transformation of the N-th-order linear difference equation into a system of the first order difference equations is presented. The proposed transformation opens possibility to obtain new forms of the N-dimensional system of the first order equations that can be useful for the analysis of solutions of the N-th-order difference equations. In particular for the third-order linear difference equation the nonlinear second-order difference equation that plays the same role as the Riccati equation for second-order linear difference equation is obtained. The new form of the Ndimensional system of first order equations can also be used to find the WKB solutions of the linear difference equation with coefficients that vary slowly with index.

On the Orthogonality of the q-Derivatives of the Discrete q-Hermite I Polynomials

2020 ◽  
pp. 135-162
Author(s):  
Sakina Alwhishi ◽  
Rezan Sevinik Adıgüzel ◽  
Mehmet Turan
Keyword(s):  
Difference Equation ◽  
Hermite Polynomials ◽  
Second Order ◽  
Orthogonality Relation ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Hypergeometric Type ◽  
Limiting Case ◽  
Derivatives Of

Discrete q-Hermite I polynomials are a member of the q-polynomials of the Hahn class. They are the polynomial solutions of a second order difference equation of hypergeometric type. These polynomials are one of the q-analogous of the Hermite polynomials. It is well known that the q-Hermite I polynomials approach the Hermite polynomials as q tends to 1. In this chapter, the orthogonality property of the discrete q-Hermite I polynomials is considered. Moreover, the orthogonality relation for the k-th order q-derivatives of the discrete q-Hermite I polynomials is obtained. Finally, it is shown that, under a suitable transformation, these relations give the corresponding relations for the Hermite polynomials in the limiting case as q goes to 1.

scholarly journals Distribution of iterates of first order difference equations

1980 ◽  
Vol 22 (1) ◽  
pp. 133-143 ◽  
Author(s):  
James B. McGuire ◽  
Colin J. Thompson
Keyword(s):  
Invariant Measure ◽  
Difference Equation ◽  
Lebesgue Measure ◽  
Difference Equations ◽  
Absolutely Continuous ◽  
Order Difference ◽  
First Order ◽  
Order Difference Equation ◽  
The Difference ◽  
Biological Context

An invariant measure which is absolutely continuous with respect to Lebesgue measure is constructed for a particular first order difference equation that has an extensive biological pedigree. In a biological context this invariant measure gives the density of the population whose growth is governed by the difference equation. Further asymptotically universal results are obtained for a class of difference equations.

scholarly journals On the Hyers-Ulam Stability of the First-Order Difference Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Soon-Mo Jung ◽  
Young Woo Nam
Keyword(s):  
Difference Equation ◽  
Ulam Stability ◽  
Order Difference ◽  
First Order ◽  
Order Difference Equation ◽  
The Difference

We prove Hyers-Ulam stability of the first-order difference equation of the formxi+1=F(i,xi), whereFis a given function with some moderate features. Moreover, we introduce some conditions for the functionFunder which the difference equation is not stable in the sense of Hyers and Ulam.

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