MR2427543 Toshiki NAITO, Pham Huu Anh NGOC and Jong Son SHIN Toshiki NAITO, Pham Huu Anh NGOC and Jong Son SHIN Funkcialaj Ekvacioj. Serio Internacia 51 2008 55--80 http://fe.math.kobe-u.ac.jp/FE/FullPapers/51-1/51_55.pdf http://www.ams.org/mathscinet-getitem?mr=MR2427543 We give a new representation of solutions of the periodic linear difference equation of the form $x(n+1)=Bx(n)+b(n)$, where $B$ is a complex $p\times p$ matrix and $b(n)\in{\mathbb C}^p$ satisfies the condition $b(n)=b(n+\rho)$, $\rho\in{\mathbb N}$, $\rho\geq 2$. If $B=e^{\tau A}$, $\tau>0$, then the equation has two representations of solutions based on $A$ and $B$. In particular, the representation of solutions based on $A$ is deduced from the one based on $B$ by using the translation formulae from $B$ to $A$. Using these representations, we can obtain the complete classification of the set of initial values according to the behavior of solutions. As applications of these results, by the initial values we characterize necessary and sufficient conditions on the existence of a bounded solution and a $\rho$-periodic solution. Periodic linear difference equation, Representation of solution, Bounded solution, Periodic solution, Asymptotic behavior of solution, Index of growth order. 39A10, 39A11. 51-55 2008 Toshiki NAITO, Pham Huu Anh NGOC and Jong Son SHIN Toshiki NAITO, Pham Huu Anh NGOC and Jong Son SHIN 1 Agarwal, R. P. Difference Equations and Inequalities, Theory, Methods, and Applications Marcel Dekker, New York-Basel-Hong Kong 1992 MR1155840 2 Elaydi, S. N. An Introduction to Difference equations Springer-Varlag, New York 1996 MR1410259

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