Note on some representations of general solutions to homogeneous linear difference equations
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 ≥ 2 $n\ge 2$ , 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. He...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-09-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-020-02944-y |
| _version_ | 1818584764175613952 |
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| author | Stevo Stević Bratislav Iričanin Witold Kosmala Zdeněk Šmarda |
| author_facet | Stevo Stević Bratislav Iričanin Witold Kosmala Zdeněk Šmarda |
| author_sort | Stevo Stević |
| collection | DOAJ |
| description | 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 ≥ 2 $n\ge 2$ , 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. |
| first_indexed | 2024-12-16T08:26:21Z |
| format | Article |
| id | doaj.art-d9df97170300443684a15b11feb9df95 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-16T08:26:21Z |
| publishDate | 2020-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-d9df97170300443684a15b11feb9df952022-12-21T22:38:00ZengSpringerOpenAdvances in Difference Equations1687-18472020-09-012020111310.1186/s13662-020-02944-yNote on some representations of general solutions to homogeneous linear difference equationsStevo Stević0Bratislav Iričanin1Witold Kosmala2Zdeněk Šmarda3Mathematical Institute of the Serbian Academy of SciencesFaculty of Electrical Engineering, Belgrade UniversityDepartment of Mathematical Sciences, Appalachian State UniversityFaculty of Electrical Engineering and Communication, Department of Mathematics, Brno University of TechnologyAbstract 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 ≥ 2 $n\ge 2$ , 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.http://link.springer.com/article/10.1186/s13662-020-02944-yHomogeneous linear difference equation with constant coefficientsGeneral solutionRepresentation of solutionsFibonacci sequence |
| spellingShingle | Stevo Stević Bratislav Iričanin Witold Kosmala Zdeněk Šmarda Note on some representations of general solutions to homogeneous linear difference equations Advances in Difference Equations Homogeneous linear difference equation with constant coefficients General solution Representation of solutions Fibonacci sequence |
| title | Note on some representations of general solutions to homogeneous linear difference equations |
| title_full | Note on some representations of general solutions to homogeneous linear difference equations |
| title_fullStr | Note on some representations of general solutions to homogeneous linear difference equations |
| title_full_unstemmed | Note on some representations of general solutions to homogeneous linear difference equations |
| title_short | Note on some representations of general solutions to homogeneous linear difference equations |
| title_sort | note on some representations of general solutions to homogeneous linear difference equations |
| topic | Homogeneous linear difference equation with constant coefficients General solution Representation of solutions Fibonacci sequence |
| url | http://link.springer.com/article/10.1186/s13662-020-02944-y |
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