THE ZECKENDORF DECOMPOSITION OF CERTAIN FIBONACCI-LUCAS PRODUCTS

The decomposition of any positive integer N as a sum of positive-subscripted, distinct, nonconsecutive Fibonacci numbers Fk is commonly referred to as the Zeckendorf decomposition ofN (ZD of N, in brief) [10]. This decomposition is always possible and, apart from the equivalent use of Fx instead of F2 (or vice-versa), is unique [8]. In the past years sequences of integers {alb}, where a and b are certain Fibonacci and/or Lucas numbers (Lk), have been investigated from the point of view of the ZD of their terms (e.g., see [3], [4], [5]). The aim of this paper is to extend these studies to sequences {ab}. More precisely, in Section 2 we establish the ZD of mFhFk and ml^^, with h and k arbitrary positive integers (possibly subject to some trivial restrictions), for the first few positive values of the integer m; the ZD of FhLk,FJ?Lk, and FhI?k are also found. In Section 3, after some brief considerations on the ZD of nFn, we analyze certain Fibonacci-Lucas products that emerge from particular choices of n. All the identities presented in this paper have been established by proving conjectures based on behavior that became apparent through the study of early cases of A, k, and n. These conjectures were made with the aid of a multi-precision program including the generation of largesubscripted Fibonacci numbers. On the other hand, once the identities were conjectured, their proofs appeared to be rather easy and similar to one another so that, to save space, we confine ourselves to proving but a few among them; this is done in Section 4. Section 5 provides a glimpse of possible further investigations. It is worth mentioning that formula (1.4) of [4], namely,