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

Advances in Difference Equations

article

Stević, Stevo¹  Iričanin, Bratislav⁴  Kosmala, Witold⁶  Šmarda, Zdeněk³ 

[1] Mathematical Institute of the Serbian Academy of Sciences;Department of Medical Research, China Medical University Hospital, China Medical University;Faculty of Electrical Engineering and Communication, Department of Mathematics, Brno University of Technology;Faculty of Electrical Engineering, Belgrade University;Faculty of Mechanical and Civil Engineering in Kraljevo, University of Kragujevac;Department of Mathematical Sciences, Appalachian State University

关键词: Homogeneous linear difference equation with constant coefficients;    General solution;    Representation of solutions;    Fibonacci sequence;   

DOI  :  10.1186/s13662-020-02944-y

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

It is known that every solution to the second-order difference equation$x_{n}=x_{n-1}+x_{n-2}=0$ ,$n\ge 2$ , can be written in the following form$x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ , where$f_{n}$ 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.

【 授权许可】

CC BY