Sums of products of two reciprocal Fibonacci numbers

Abstract

In this paper, we employ elementary methods to investigate the reciprocal sums of the products of two Fibonacci numbers in several ways. First, we consider the sums of the reciprocals of the products of two Fibonacci numbers and establish five interesting families of identities. Then we extend such analysis to the alternating sums and obtain five analogous results. MSC: 11B39

Asymptotic behavior of reciprocal sum of two products of Fibonacci numbers

“…for any m ∈ N and 0 ≤ ≤ m -1. One can see the results for the product of two Fibonacci numbers in [5].…”

Some results on the infinite sums of reciprocal generalized Fibonacci numbers

“…Then 7follows from (9) and (10). (b) If Φ > 0, then, from the proof of (a), it is easily seen that there exists a positive integer m 2 such that…”

“…Following the work of Ohtsuka and Nakamura, diverse results in the same direction have been reported in the literature [1][2][3][4][5], [7][8][9][10], [13], [14]. In particular, Wang and Zhang [14] considered the reciprocal sums of even-indexed and odd-indexed Fibonacci numbers, and obtained Theorem 1.2 below.…”

On the finite sums of products of reciprocal Fibonacci and Lucas numbers

“…In recent years, much attention was given to the properties related to the reciprocal sums of Fibonacci or other numbers [1][2][3][4][5], [7][8][9][10]. Ohtsuka and Nakamura [8] found interesting properties of the Fibonacci numbers and proved Theorem 1.1 below, where • indicates the floor function and N e (N o , respectively) denotes the set of positive even (odd, respectively) integers.…”