On second-order linear recurrent functions with period k and proofs to two conjectures of Sroysang

Let w be a real-valued function on R and k be a positive integer. If for every real number x, winsert ignore into journalissuearticles values(x + 2k); = rwinsert ignore into journalissuearticles values(x + k); + swinsert ignore into journalissuearticles values(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 winsert ignore into journalissuearticles values(x + 2k); = −rwinsert ignore into journalissuearticles values(x + k); + swinsert ignore into journalissuearticles values(x); an odd secondorder linear recurrent function with period k. In this work, 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 [9]. 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 [19]. 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 [9] and contribute new results in the development of this topic.