On a Higher-Order Difference Equation

We describe in an elegant and short way the behaviour of positive solutions of the higher-order difference equation , , where and , extending some recent results in the literature.


Introduction
Studying difference equations has attracted a considerable interest recently, see, for example, 1-39 and the references listed therein.The study of positive solutions of the following higher-order difference equations: , n ∈ N 0 , 1.1 and where A, B > 0, p i , q i are natural numbers such that Case A 0 is of some less interest, since in this case positive solutions of 1.1 and 1.2 , by using the change y n ln x n , become solutions of a linear difference equation with constant coefficients.However, some particular results for the case recently appeared in the literature, see 16, 17, 39 .Nevertheless, motivated by the above-mentioned papers, we will describe the behaviour of positive solutions of the higher-order difference equation where p, q ∈ N and c > 0, in, let us say, an elegant and short way.
Let us introduce the following.
3 is said to be eventually periodic with period τ if there is n 0 ∈ {− p q , . . ., −1, 0, 1, . ..} such that x n τ x n for all n ≥ n 0 .If n 0 − p q , then we say that the sequence x n ∞ n − p q is periodic with period τ.
For some results on equations all solutions of which are eventually periodic see, for example, 2, 4, 8, 15, 28, 37 and the references therein. 1.4 Now we return to 1.3 .First, note that if p q, then 1.3 becomes from which easily follow the following results: a if c 1, then all positive solutions of 1.5 are periodic with period 2p; b if c ∈ 0, 1 , then each positive solution of 1.5 converges to zero.Moreover, its subsequences x 2pm−i m∈N 0 , i 1, 2, . . ., 2p, converges decreasingly to zero as m → ∞; Moreover, its subsequences x 2pm−i m∈N 0 , i 1, 2, . . ., 2p, tend increasingly to infinity as m → ∞.
We may assume that p and q are relatively prime integers, that is, gcd p, q 1 the greatest common divisor of numbers p and q .Namely, if gcd p, q r > 1, then by using the changes z i m x mr i , i 0, 1, . . ., r − 1, 1.3 is reduced to r copies of the following equation: where p 1 p/r, q 1 q/r, c > 0, and gcd p 1 , q 1 1.
Further, note that from 1.3 , we have that which implies that the sequence u n x n x n−q , n ≥ −p, satisfies the following simple difference equation:

Main Results
Here we formulate and prove our main results.
Theorem 2.1.Assume that c 1, gcd p, q 1, and p is odd, then all positive solutions of 1.3 are eventually periodic with period τ 2pq.
Proof.By using repeatedly relation 1.7 p-times, we obtain Now, note that from 1.8 , it follows that in this case u n is periodic with period p.On the other hand, since gcd p, q 1 for each i, j ∈ {0, 1, . . ., p − 1}, i / j, we have that n − 2i 1 q − n − 2j 1 q j − i 2q / ≡ 0 mod p , n − 2i 2 q − n − 2j 2 q j − i 2q / ≡ 0 mod p .
By taking the logarithm of 1.3 and using the change v n ln x n , we get The characteristic polynomial of the homogeneous part of 2.4 is from which it follows that all its roots are expressed by exp 2k 1 πi q , k 0, 1, . . ., q − 1, exp 2lπi p , l 0, 1, . . ., p − 1.

2.6
These roots are simple if and only if Clearly, if p is odd, inequality 2.7 holds.If p is even, that is, p 2 s r, for some s, r ∈ N, then, since gcd p, q 1, q must be odd.Then, for k q − 1 /2 and l r, we will get that inequality 2.7 does not hold.
From the above consideration and Theorem 2.

2.9
Note that from 2.9 , it follows that x n c n/2p x n , 2.10 and that x n is a positive solution of 1.3 with c 1. From 2.9 , 2.10 , and Theorem 2.1 the following results directly follow.
Finally, there are two concluding interesting remarks.
Remark 2.5.Note that, since the functions cos 2k 1 πn/q and sin 2k 1 πn/q are periodic with period 2q and the functions cos 2lπn/p and sin 2lπn/p are periodic with period p, from the representation 2.9 we also obtain Theorem 2.1.
Remark 2.6.The results in papers 16, 17, 39 , which are obtained in much complicated ways, are particular cases of our results.Namely, in 16 Özban studied a system which is transformed into 1.3 with p 1, q m k 1 and c 1, in 17 he studied a system which is transformed into 1.3 with p 3, and c b/a, while in 39 the authors considered a system which is transformed into 1.3 with c b/a, but they only considered the case when p ≤ q.
1, we get the next corollary.Then all positive solutions of 1.3 are eventually periodic if and only if p is odd.Moreover, if p is odd, then the period is τ 2pq.