On Reciprocal Series of Generalized Fibonacci Numbers with Subscripts in Arithmetic Progression

Copyright q 2012 Neşe Ömür. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate formulas for closely related series of the forms: ∑∞ n 01/ Uan b c , ∑∞ n 0 −1 Uan b/ Uan b c , ∑∞ n 0U2 an b / U 2 an b c 2 for certain values of a, b, and c.


Introduction
Let p be a nonzero integer such that Δ p 2 4 / 0. The generalized Fibonacci and Lucas sequences are defined by the following recurrences: where U 0 0, U 1 1 and V 0 2, V 1 p, respectively.When p 1, U n F n nth Fibonacci number and V n L n nth Lucas number .
If α and β are the roots of equation x 2 − px − 1 0, the Binet formulas of the sequences {U n } and {V n } have the forms: In 1 , Backstrom developed formulas for closely related series of the form: Discrete Dynamics in Nature and Society for certain values of a, b, and c.For example, he obtained the following series: where K represents an odd integer and t is an integer in the range − K − 1 /2 to K − 1 /2 inclusive.Also, he gave the similar results for Lucas numbers.In 2 , Popov found in explicit form series of the form: for certain values of a, b, c, and d.
In 3 , Popov generalized some formulas of Backstrom 1 related to sums of reciprocal series of Fibonacci and Lucas numbers.For example, where s and r are integers.In 4 , Gauthier found the closed form expressions for the following sums: where for x / 0 an indeterminate, the generalized Fibonacci and Lucas polynomials {f n } n and {l n } n are given by the following recurrences: In this paper, we investigate formulas for closely related series of the forms: for certain values of a, b and c.

On Some Series of Reciprocals of Generalized Fibonacci Numbers
In this section, firstly, we will give the following lemmas for further use.
Lemma 2.1.Let n be an arbitrary nonzero integer.For integer m ≥ 1, and for integer m ≥ 0, Proof.We give the proof of Lemma 2.1 as the proofs of the sums in 4 , using the following equalities:

2.5
Proof.From Binet formulas of sequences {U n } and {V n }, the desired results are obtained.

2.6
Proof.By replacing n with 2n 1 t in 2.5 , we have Taking r 2n 1 t and s t in the equality V s U r U r s −1 s U r−s 5 , the equality 2.8 is rewritten as follows: We have the sum For an odd integer t, we have and taking s 2nt and r 2t in identity 5 : Substituting 2.11 and 2.13 in 2.10 , we have the desired result.
For example, if we take t 1 and p 1 in 2.6 , we have