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).