#### Abstract

We investigate formulas for closely related series of the forms: , , for certain values of , , and .

#### 1. Introduction

Let be a nonzero integer such that . The generalized Fibonacci and Lucas sequences are defined by the following recurrences: where , and ,, respectively. When , (th Fibonacci number) and (th Lucas number).

If and are the roots of equation , the Binet formulas of the sequences and have the forms: respectively.

In [1], Backstrom developed formulas for closely related series of the form: for certain values of , , and . For example, he obtained the following series: where represents an odd integer and is an integer in the range to 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 , , , and .

In [3], Popov generalized some formulas of Backstrom [1] related to sums of reciprocal series of Fibonacci and Lucas numbers. For example, where and are integers.

In [4], Gauthier found the closed form expressions for the following sums: where for an indeterminate, the generalized Fibonacci and Lucas polynomials and are given by the following recurrences: respectively.

In this paper, we investigate formulas for closely related series of the forms: for certain values of , and .

#### 2. 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 be an arbitrary nonzero integer. For integer ,
**
and for integer ,
*

*Proof. *We give the proof of Lemma 2.1 as the proofs of the sums in [4], using the following equalities:

Lemma 2.2. *For arbitrary integers and ,
*

*Proof. *From Binet formulas of sequences and , the desired results are obtained.

Theorem 2.3. *For an odd integer ,
*

*Proof. *By replacing with in (2.5), we have
or
Taking and in the equality [5], the equality (2.8) is rewritten as follows:
We have the sum
For an odd integer , we have
and taking and in identity [5]:
we get

Substituting (2.11) and (2.13) in (2.10), we have the desired result.

For example, if we take and in (2.6), we have Note that

Corollary 2.4. *For an odd integer ,
*

*Proof. *Using the equalities and in Theorem 2.3, the results are obtained.

Corollary 2.5. *Let be an odd integer. For and ,
**
and for and ,
*

*Proof. *Since
the results are easily seen by equalities (2.16).

Theorem 2.6. *For an integer and an arbitrary nonzero integer ,
*

*Proof. *By replacing with and with in (2.4), we have
or
Multiplying equality (2.22) by , we get
We have the sum:
Using the equalities (2.1) and (2.21), the proof is obtained.

Corollary 2.7. *For an arbitrary nonzero integer ,
*

*Proof. *Taking in Theorem 2.6 and using (2.19), the result is easily obtained.

Theorem 2.8. *For an integer and an arbitrary nonzero integer ,
*

*Proof. *The proof of the theorem is similar to the proof of Theorem 2.6.

Corollary 2.9. *For an arbitrary nonzero integer ,
*

*Proof. *Taking in Theorem 2.8 and using (2.19), the result is easily obtained.

For example, if we take and in (2.27), we have

Theorem 2.10. *For an integer and an arbitrary nonzero integer ,
*

*Proof. *The proof of theorem is similar to the proof of Theorem 2.6.

Corollary 2.11. *For an arbitrary nonzero integer ,
*

*Proof. *Taking in Theorem 2.10 and using (2.19), the result is easily obtained.

For example, if we take in the equality (2.30), we have