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

Year 2016, Volume: 45 Issue: 2, 429 - 446, 01.04.2016

Julius Fergy Tiongson Rabago

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 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.

Keywords

Second-order linear recurrent functions with period k, Horadam numbers, generalized Fibonacci sequences, Pell functions with period k, Jacobsthal functions with period k, Sroysang’s conjecture

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