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Discrete Dynamics in Nature and Society
VolumeΒ 2012, Article IDΒ 684280, 7 pages
http://dx.doi.org/10.1155/2012/684280
Research Article

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

Department of Mathematics, Kocaeli University, 41380 Izmit, Turkey

Received 24 October 2011; Accepted 16 November 2011

Academic Editor: ElenaΒ Braverman

Copyright Β© 2012 Neşe Γ–mΓΌr. 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.

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

References

  1. R. P. Backstrom, β€œOn reciprocal series related to Fibonacci numbers with subscripts in arithmetic progression,” The Fibonacci Quarterly, vol. 19, pp. 14–21, 1981. View at Google Scholar Β· View at Zentralblatt MATH
  2. B. S. Popov, β€œOn certain series of reciprocals of Fibonacci numbers,” The Fibonacci Quarterly, vol. 22, no. 3, pp. 261–265, 1984. View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  3. B. S. Popov, β€œSummation of reciprocal series of numerical functions of second order,” The Fibonacci Quarterly, vol. 24, no. 1, pp. 17–21, 1986. View at Google Scholar Β· View at Zentralblatt MATH
  4. N. Gauthier, β€œSolution to problem H-680,” The Fibonacci Quarterly, vol. 49, no. 1, pp. 90–92, 2011. View at Google Scholar
  5. S. Vajda, Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications, John Wiley & Sons, New York, NY, USA, 1989.