Dynamical analysis of discrete time equations with a generalized order

On a Study of a Family of Higher-Order Recurrence Relations

“…These invariants are utilized to derive the solutions in closed form. We demonstrated that all the formula solutions in [8] are special cases of our findings. Some conditions for existence of 4k and 8k periodic solutions were established.…”

“…We plot Figure 1 to illustrate Theorem 4.1. We note that for A = −1 and B = 1, we get the result in Theorem 13 in [8] and the result's restriction ( k−1 r=0 η −j+4 j 4 −4r = 2, j = 0, 1, . .…”

“…, 4k−1. Replacing b = ±1 in (42) yields the results in [8] (see Theorems 1 and 6). In fact, noting that τ (4k − 1 − j) = 3 − τ (j) = 3 − j + 4 j/4 , we have:…”

“…In the following section, we investigate some special cases. One of the aims is to realize some results in [8].…”

“…Examples of the use of difference equations in real-life include modeling the proliferation of disease, loan payments, population studies, etc. Recurrence equations of a general order have been investigated in the literature and from different approaches by several authors [3][4][5][6][7][8][9][10][11][12]. In their work, Almatrafi and others [8] discussed the exact solutions, stability, oscillation and periodic aspects of the the difference equations…”