Abstract and Applied Analysis
Volume 2014 (2014), Article ID 402540, 4 pages
http://dx.doi.org/10.1155/2014/402540

Research Article

On the Sum of Reciprocal Generalized Fibonacci Numbers

Pingzhi Yuan, Zilong He, and Junyi Zhou

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

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

Academic Editor: Antonio M. Peralta

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 [1–9]. 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

S. H. Holliday and T. Komatsu, “On the sum of reciprocal generalized Fibonacci numbers,” Integers, vol. 11, no. 4, pp. 441–455, 2011.

E. Kılıç and T. Arıkan, “More on the infinite sum of reciprocal Fibonacci, Pell and higher order recurrences,” Applied Mathematics and Computation, vol. 219, no. 14, pp. 7783–7788, 2013.

T. Komatsu, “On the nearest integer of the sum of r eciprocal Fibonacci numbers, A-portaciones,” Matematicas Investigacion, vol. 20, pp. 171–184, 2011.

T. Komatsu and V. Laohakosol, “On the sum of reciprocals of numbers satisfying a recurrence relation of order s,” Journal of Integer Sequences, vol. 13, no. 5, Article ID 10.5.8, pp. 1–9, 2010.

H. Ohtsuka and S. Nakamura, “On the sum of reciprocal Fibonacci numbers,” The Fibonacci Quarterly, vol. 46-47, no. 2, pp. 153–159, 2008-2009.

R. Ma and W. Zhang, “Several identities involving the Fibonacci numbers and Lucas numbers,” The Fibonacci Quarterly, vol. 45, no. 2, pp. 164–170, 2007.

Z. Xu and T. Wang, “The infinite sum of the cubes of reciprocal Pell numbers,” Advances in Difference Equations, vol. 2013, article 184, 2013.

Z. Wenpeng and W. Tingting, “The infinite sum of reciprocal Pell numbers,” Applied Mathematics and Computation, vol. 218, no. 10, pp. 6164–6167, 2012.

W. Zhang and T. Wang, “The infinite sum of reciprocal of the square of the Pell numbers,” Journal of Weinan Teacher's University, vol. 26, pp. 39–42, 2011.