Authors: Stević, Stevo 
Iričanin, Bratislav
Kosmala, Witold
Šmarda, Zdeněk
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Note on some representations of general solutions to homogeneous linear difference equations
Journal: Advances in Difference Equations
Volume: 2020
Issue: 1
First page: Article no. 486
Issue Date: 10-Sep-2020
Rank: M21a
ISSN: 1687-1847
DOI: 10.1186/s13662-020-02944-y
Abstract: 
It is known that every solution to the second-order difference equation xn= xn−1+ xn−2= 0 , n≥ 2 , can be written in the following form xn= xfn−1+ x1fn, 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.
Keywords: Fibonacci sequence | General solution | Homogeneous linear difference equation with constant coefficients | Representation of solutions
Publisher: Springer Link

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