ON SECOND-ORDER LINEAR RECURRENT FUNCTIONS WITH PERIOD k AND PROOFS TO TWO CONJECTURES OF SROYSANG

Abstract

Let w be a real-valued function on R and k be a positive integer. If for every real number x, w(x + 2k) = rw(x + k) + sw(x) for some nonnegative real numbers r and s, then we call such function a second-order linear recurrent function with period k. Similarly, we call a function w : R → R satisfying w(x + 2k) = −rw(x + k) + sw(x) an odd secondorder linear recurrent function with period k. In the present paper, we present some elementary properties of these type of functions and develop the concept using the notion of f -even and f -odd functions discussed in [7]. We also investigate the products and quotients of these functions and provide in this work a proof of the conjecture of B. Sroysang which he posed in [13]. In fact, we offer here a proof of a more general case of the problem. Consequently, we present findings that confirm recent results in the theory of Fibonacci functions ([7]) and contribute new results in the development of this topic.

On Third-Order Linear Recurrent Functions

“…Here we get results obtained in [10] with third-order linear recurrence. We give the limit of the quotient of a Tribonacci function with period , extending the results of [11] in third-order linear recurrence.…”

On second-order linear recurrent homogeneous differential equations with period

“…, is a solution to an odd Fibonacci-like homogeneous differential equation with period 3. i.e., f (x) = e tx is a solution to (2.12) f (6) Then,…”

“…In particular, Sroysang defined a function f (x) : R → R as a Fibonacci function of period k, (k ∈ N) if it satisfies the equation f (x + 2k) = f (x + k) + f (x) for all x ∈ R. Recently, the notion of Fibonacci function has been further generalized by the author in [6]. The concept of second-order linear recurrent functions with period k which has been introduced by the author in [6] gave rise to the concept of Pell and Jacobsthal functions with period k, which are analogues of Fibonacci functions. Some elementary properties of these newly defined functions were also presented by the author in [6].…”

Well-Defined Solutions of a Three-Dimensional System of Difference Equations

“…Difference equations and their systems are related to many real life models in different branches of modern science such as biology, physics, economics, etc, in [1,2]. This is due to the fact that these models are expressed by discrete equations and this explain why difference equations and their systems have attracted the attention of many researchers in recent years in [3][4][5][6]. One of the most popular subject associated with difference equations and their system, especially the non-linear ones, is to examine their solvability and the behavior of their solutions in .…”