On the Sum of Reciprocal Generalized Fibonacci Numbers

Abstract

The Fibonacci Zeta functions are defined by F .s/ D P 1 kD1 F s k . Several aspects of the function have been studied. In this article we generalize the results by Ohtsuka and Nakamura, who treated the partial infinite sum P 1 kDn F s k for all positive integers n.

Identities Involving the Fourth-Order Linear Recurrence Sequence

“…In particular, for x = 1, F n = F n (x) are the famous Fibonacci numbers. These polynomials and numbers play extremely vital roles in the mathematical theories and applications and a significant amount of research has been carried out to obtain a variety of meaningful results by several authors (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]). For example, Yuan Yi and Wenpeng Zhang (see [16]) researched the computational problem of the summation: ∑ a 1 +a 2 +···+a h+1 =n F a 1 (x)F a 2 (x) · · · F a h+1 (x).…”

Some results on the infinite sums of reciprocal generalized Fibonacci numbers

“…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.…”

“…and (6) are obtained from (8) and 13, respectively. Also it can be shown that Corollaries presented in [4] and [9]…”

On the reciprocal sums of products of Fibonacci and Lucas numbers

“…Following the work of Ohtsuka and Nakamura [8], diverse results in the same direction have appeared in the literature [1], [3][4][5], [9][10][11][12]. In particular, according to Holliday and Komatsu [5], the infinite sums of reciprocal Lucas numbers satisfy the identities given in Theorem 1.2 below. Theorem 1.2.…”