scholarly journals On the integration of the homogeneous linear difference equation of second order

1909 ◽  
Vol 10 (3) ◽  
pp. 319-319 ◽  
Author(s):  
Walter B. Ford
Keyword(s):  
Difference Equation ◽  
Linear Difference Equation ◽  
Second Order ◽  
Homogeneous Linear

scholarly journals Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation

2017 ◽  
Vol 19 (06) ◽  
pp. 1650056 ◽  
Author(s):  
Carlos E. Arreche
Keyword(s):  
Difference Equation ◽  
Galois Group ◽  
Galois Theory ◽  
Rational Functions ◽  
Linear Difference Equation ◽  
Second Order ◽  
Differential Galois Group ◽  
The Difference ◽  
Differential Relations ◽  
Homogeneous Linear

We apply the difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous linear difference equation [Formula: see text] where the coefficients [Formula: see text] are rational functions in [Formula: see text] with coefficients in [Formula: see text]. We develop algorithms to compute the difference-differential Galois group associated to such an equation, and show how to deduce the differential-algebraic relations among the solutions from the defining equations of the Galois group.

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

scholarly journals Solution and Interpretation of Neutrosophic Homogeneous Difference Equation

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1091
Author(s):  
Abdul Alamin ◽  
Sankar Prasad Mondal ◽  
Shariful Alam ◽  
Ali Ahmadian ◽  
Soheil Salahshour ◽  
...  
Keyword(s):  
Difference Equation ◽  
Discrete System ◽  
Characterization Theorem ◽  
Theoretical Work ◽  
Linear Difference Equation ◽  
Neutrosophic Set ◽  
Actuarial Science ◽  
Initial Information ◽  
Information Coefficient ◽  
Homogeneous Linear

In this manuscript, we focus on the brief study of finding the solution to and analyzingthe homogeneous linear difference equation in a neutrosophic environment, i.e., we interpreted the solution of the homogeneous difference equation with initial information, coefficient and both as a neutrosophic number. The idea for solving and analyzing the above using the characterization theorem is demonstrated. The whole theoretical work is followed by numerical examples and an application in actuarial science, which shows the great impact of neutrosophic set theory in mathematical modeling in a discrete system for better understanding the behavior of the system in an elegant manner. It is worthy to mention that symmetry measure of the systems is employed here, which shows important results in neutrosophic arena application in a discrete system.

Fuzzy Difference Equations: The Initial Value Problem

2001 ◽  
Vol 5 (6) ◽  
pp. 315-325 ◽  
Author(s):  
James J. Buckley ◽  
◽  
Thomas Feuring ◽  
Yoichi Hayashi ◽  
◽  
...  
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
National Income ◽  
Linear Difference Equation ◽  
Second Order ◽  
Solution Methods ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
The Difference

In this paper we study fuzzy solutions to the second order, linear, difference equation with constant coefficients but having fuzzy initial conditions. We look at two methods of solution: (1) in the first method we fuzzify the crisp solution and then check to see if it solves the difference equation; and (2) in the second method we first solve the fuzzy difference equation and then check to see if the solution defines a fuzzy number. Relationships between these two solution methods are also presented. Two applications are given: (1) the first is about a second order difference equation, having fuzzy initial conditions, modeling national income; and (2) the second is from information theory modeling the transmission of information.

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 Solution of a general homogeneous linear difference equation

1980 ◽  
Vol 22 (2) ◽  
pp. 254-254
Author(s):  
V. N. Singh
Keyword(s):  
Difference Equation ◽  
Linear Difference Equation ◽  
Image Position ◽  
Homogeneous Linear

In Section 3, since relation (6) is valid only for n ≥ 2r, the condition n ≥ r in relation (9) should be replaced by n ≥ 2r. When r ≤ n ≤ 2r − 1, relation (9) still holds but now relations (6) and (7) should be replaced, respectively, by the relationsandthe asterisk again denoting omission of the last column. The proof of (9) for r ≤ n ≤ 2r − 1 is exactly similar to its proof for n ≥ 2r.

scholarly journals Solution of a general homogeneous linear difference equation

1980 ◽  
Vol 22 (1) ◽  
pp. 53-57 ◽  
Author(s):  
V. N. Singh
Keyword(s):  
Difference Equation ◽  
Linear Difference Equation ◽  
The Other ◽  
Term Linear ◽  
Homogeneous Linear

AbstractSolutions of a homogeneous (r + 1)-term linear difference equation are given in two different forms. One involves the elements of a certain matrix, while the other is in terms of certain lower Hessenberg determinants. The results generalize some earlier results of Brown [1] for the solution of a 3-term linear difference equation.

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