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General solutions to systems of difference equations and some of their representations

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Abstract

Here we solve the following system of difference equations

$$ x^{(j)}_{n+1}=\frac{F_{m+2}+F_{m+1}x^{((j+1)mod(p))}_{n-k}}{F_{m+3} +F_{m+2}x^{((j+1)mod(p))}_{n-k}},\quad n,m, p, k \in N_0, j=\overline{1,p}, $$

where \(\left( F_{n}\right) _{n=0}^{+\infty }\) is the Fibonacci sequence. We give a representation of its general solution in terms of Fibonacci numbers and the initial values. Some theoretical justifications related to the representation for the general solution are also given.

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Acknowledgements

This work was supported by Directorate General for Scientific Research and Technological Development (DGRSDT), Algeria.

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Correspondence to Yacine Halim.

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Khelifa, A., Halim, Y. General solutions to systems of difference equations and some of their representations. J. Appl. Math. Comput. 67, 439–453 (2021). https://doi.org/10.1007/s12190-020-01476-8

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  • DOI: https://doi.org/10.1007/s12190-020-01476-8

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