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

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dc.contributor.authorStevič, Stevocs
dc.contributor.authorIričanin, Bratislavcs
dc.contributor.authorKosmala, Witoldcs
dc.contributor.authorŠmarda, Zdeněkcs
dc.coverage.issue1cs
dc.coverage.volume2020cs
dc.date.accessioned2021-04-22T10:54:12Z
dc.date.available2021-04-22T10:54:12Z
dc.date.issued2020-09-10cs
dc.description.abstractIt is known that every solution to the second-order difference equation x(n) = x(n-1) + x(n-2) = 0, n >= 2, can be written in the following form x(n) = x(0)f(n-1) + x(1)f(n), where fn 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.en
dc.formattextcs
dc.format.extent1-13cs
dc.format.mimetypeapplication/pdfcs
dc.identifier.citationAdvances in Difference Equations. 2020, vol. 2020, issue 1, p. 1-13.en
dc.identifier.doi10.1186/s13662-020-02944-ycs
dc.identifier.issn1687-1847cs
dc.identifier.other165050cs
dc.identifier.urihttp://hdl.handle.net/11012/196552
dc.language.isoencs
dc.publisherSpringer Naturecs
dc.relation.ispartofAdvances in Difference Equationscs
dc.relation.urihttps://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-020-02944-ycs
dc.rightsCreative Commons Attribution 4.0 Internationalcs
dc.rights.accessopenAccesscs
dc.rights.sherpahttp://www.sherpa.ac.uk/romeo/issn/1687-1847/cs
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/cs
dc.subjectHomogeneous linear difference equation with constant coefficientsen
dc.subjectGeneral solutionen
dc.subjectRepresentation of solutionsen
dc.subjectFibonacci sequenceen
dc.titleNote on some representations of general solutions to homogeneous linear difference equationsen
dc.type.driverarticleen
dc.type.statusPeer-revieweden
dc.type.versionpublishedVersionen
sync.item.dbidVAV-165050en
sync.item.dbtypeVAVen
sync.item.insts2021.04.22 12:54:12en
sync.item.modts2021.04.22 12:14:34en
thesis.grantorVysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií. Ústav matematikycs
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