scholarly journals On the Solution of a Second Order Linear Homogeneous Difference Equation with Variable Coefficients

1997 ◽  
Vol 215 (1) ◽  
pp. 32-47 ◽  
Author(s):  
Ranjan K Mallik
Keyword(s):  
Difference Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Order Linear

Difference equations, determinants and the Secret Santa problem

2005 ◽  
Vol 89 (514) ◽  
pp. 2-6
Author(s):  
Tony Ward
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Linear Difference Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Auxiliary Equation ◽  
Order Linear ◽  
Particular Solution

In [1] it is shown that every second-order linear difference equation with, in general, variable coefficients can be reduced to the formIt is also shown that (1) can be solved explicitly provided that a particular solution a (n) of the ‘auxiliary equation’can be found, and many cases in which a solution of (2) is available are discussed. However, [1] is defective in that no method of finding a solution of (2) in the general case is given.

Fibonacci and Lucas polynomials

1981 ◽  
Vol 90 (3) ◽  
pp. 385-387 ◽  
Author(s):  
B. G. S. Doman ◽  
J. K. Williams
Keyword(s):  
Difference Equation ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Linear Difference Equation ◽  
Second Order ◽  
Lucas Numbers ◽  
Order Linear ◽  
Lucas Polynomials ◽  
Fibonacci And Lucas Numbers

The Fibonacci and Lucas polynomials Fn(z) and Ln(z) are denned. These reduce to the familiar Fibonacci and Lucas numbers when z = 1. The polynomials are shown to satisfy a second order linear difference equation. Generating functions are derived, and also various simple identities, and relations with hypergeometric functions, Gegenbauer and Chebyshev polynomials.

scholarly journals Instability of second-order nonhomogeneous linear difference equations with real-valued coefficients

2021 ◽  
Vol 37 (3) ◽  
pp. 489-495
Author(s):  
MASAKAZU ONITSUKA ◽  
◽  
Keyword(s):  
Difference Equation ◽  
Linear Difference Equation ◽  
Second Order ◽  
Ulam Stability ◽  
Necessary And Sufficient Condition ◽  
Linear Difference Equations ◽  
Real Numbers ◽  
Step Size ◽  
Order Linear ◽  
Necessary And Sufficient

In J. Comput. Anal. Appl. (2020), pp. 152--165, the author dealt with Hyers--Ulam stability of the second-order linear difference equation $\Delta_h^2x(t)+\alpha \Delta_hx(t)+\beta x(t) = f(t)$ on $h\mathbb{Z}$, where $\Delta_hx(t) = (x(t+h)-x(t))/h$ and $h\mathbb{Z} = \{hk|\,k\in\mathbb{Z}\}$ for the step size $h>0$; $\alpha$ and $\beta$ are real numbers; $f(t)$ is a real-valued function on $h\mathbb{Z}$. The purpose of this paper is to clarify that the second-order linear difference equation has no Hyers--Ulam stability when the step size $h>0$ and the coefficients $\alpha$ and $\beta$ satisfy suitable conditions. Finally, a necessary and sufficient condition for Hyers--Ulam stability is obtained.

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