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`Abstract and Applied AnalysisVolume 2014, Article ID 402540, 4 pageshttp://dx.doi.org/10.1155/2014/402540`
Research Article

On the Sum of Reciprocal Generalized Fibonacci Numbers

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received 19 August 2014; Accepted 26 November 2014; Published 10 December 2014

Copyright © 2014 Pingzhi Yuan et al. 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 consider infinite sums derived from the reciprocals of the generalized Fibonacci numbers. We obtain some new and interesting identities for the generalized Fibonacci numbers.

1. Introduction

For any integer , the famous Fibonacci numbers and Pell numbers are defined by the second-order linear recurrence sequences There are many interesting results on the properties of these two sequences; see [19]. In 2009, Ohtsuka and Nakamura [5] studied the properties of the Fibonacci numbers and proved the following two interesting identities: where is the floor function; that is, it denotes the greatest integer less than or equal to . Recently, Holliday and Komatsu [1] (Theorems 3 and 4) and Xu and Wang [7] proved the following interesting identities for the Pell numbers: where providing. In [7, 8], the authors asked whether there exists a computational formula for where is a positive integer.

Let and be integers such that . Define the generalized Fibonacci sequence , briefly , as shown: for where , . The Binet formula for is where .

The main purpose of this paper related to the computing problem of for and . For easy computation, we assume that is a positive integer and throughout the paper. We have the following.

Theorem 1. Let be a positive integer, and let be defined by the second-order linear recurrence sequence . Then for all one has

2. Proof of the Main Result

In this section, we will prove our main result. We consider the case that and .

Proof. From the Taylor series expansion of as , we have Using (6), we have
It is easy to check that holds for and .
Thus where Since , we have holds for and .
Taking reciprocal, we get where since An easy calculation shows that holds for and . Therefore, where for and .
Similarly, we have Since and for and , we have , whence we can take for and .
Consequently, we have shown that where for and , and for and .
Now the calculations show that
The calculations also show that for and ; for and ; and for and ; for and . Combining the calculations and (23), we obtain
Therefore we have proved Theorem 1.

Remark 2. We can also compute the cases or ; however, the computations are much more complicated. So we stop here.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

P. Yuan's research is supported by the NSF of China (Grant no. 11271142) and the Guangdong Provincial Natural Science Foundation (Grant no. S2012010009942).

References

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