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The sums of the reciprocals of Fibonacci polynomials and Lucas polynomials
Journal of Inequalities and Applications volume 2012, Article number: 134 (2012)
Abstract
In this article, we consider infinite sums derived from the reciprocals of the Fibonacci polynomials and Lucas polynomials, and infinite sums derived from the reciprocals of the square of the Fibonacci polynomials and Lucas polynomials. Then applying the floor function to these sums, we obtain several new equalities involving the Fibonacci polynomials and Lucas polynomials.
Mathematics Subject Classification (2010): Primary, 11B39.
1. Introduction
For any variable quantity x, the Fibonacci polynomials F n (x) and Lucas polynomials L n (x) are defined by Fn+2(x) = xFn+1(x) + F n (x), n ≥ 0 with the initial values F0(x) = 0 and F1(x) = 1; Ln+2(x) = xLn+1(x) + L n (x), n ≥ 0 with the initial values L0(x) = 2 and L1(x) = x. For x = 1 we obtain the usual Fibonacci numbers and Lucas numbers. Let and , then from the properties of the second-order linear recurrence sequences we have
Various authors studied the properties of Fibonacci polynomials and Lucas polynomials, and obtained many interesting results, see [1–3]. Recently, several authors studied the infinite sums derived from the reciprocals of the Fibonacci numbers and Pell numbers, and obtained some important results. For example, Ohtsuka and Nakamura [4] studied the properties of the Fibonacci numbers, and proved the following conclusions:
Wenpeng and Tingting [5] studied the infinite sums derived from the Pell numbers, and obtained two similar results.
In this article, we considered infinite sums derived from the reciprocal of the Fibonacci polynomials and Lucas polynomials, and proved the following:
Theorem 1. For any positive integer x, we have
Theorem 2. For any positive integer x, we have
Theorem 3. For any positive integer x, we have
Theorem 4. For any positive integer x ≥ 2, we have
If x = 1 and x = 2, then from our theorems we can deduce the conclusions of [4, 5]. Especially, we also have the following:
Corollary 1. For any positive integer n, we have the identities
Corollary 2. For any positive integer n, we have the identities
Corollary 3. For any positive integer n, we have the identities
2. Proof of theorems
In this section, we shall complete the proof of our theorems. We shall prove only Theorems 1 and 2, and the other two theorems are proved similarly and omitted. First we prove Theorem 1. We consider the case that n = 2m is even. At this time, Theorem 1 equivalent to
Now we prove that for any positive integers x and k ≥ 1,
This inequality equivalent to
or
applying the identities
the inequality (2.2) equivalent to
or
It is easy to check that the inequality (2.3) holds for any positive integers x and k ≥ 1. So the inequality (2.2) is true. Using (2.2) repeatedly, we have
On the other hand, we prove that for any positive integers x and k ≥ 1,
The inequality (2.5) equivalent to
or
It is easy to check that the inequality (2.6) holds for all positive integers x and k ≥ 1. So the inequality (2.5) is true. Using (2.5) repeatedly, we have
Now the inequality (2.1) follows from (2.4) and (2.7).
Similarly, we can consider the case that n = 2m + 1 is odd. Note that F1(x) - F0(x) - 1 = 0 and
So Theorem 1 is true if m = 0. If m ≥ 1, then our Theorem 1 equivalent to the inequality
First we can prove that for any positive integers x and k ≥ 1,
The inequality (2.9) equivalent to
or
It is easy to check that the inequality (2.10) holds for all positive integers x and k ≥ 1. So the inequality (2.9) is true. Using (2.9) repeatedly, we have
On the other hand, we prove that for any positive integers x and k ≥ 1,
The inequality (2.12) equivalent to
or
It is easy to check that the inequality (2.13) holds for all positive integers x and k ≥ 1. So the inequality (2.12) is true. Using (2.12) repeatedly, we have
Combining (2.11) and (2.14) we deduce the inequality (2.8). This proves Theorem 1.
Proof of Theorem 2. First we consider the case that n = 2m is even. At this time, Theorem 2 equivalent to
Now we prove that for any positive integers x and k ≥ 1,
So the inequality (2.16) equivalent to
or
It is clear that the inequality (2.17) holds for all positive integers x and k ≥ 1. So the inequality (2.16) is true. Using (2.16) repeatedly, we have
On the other hand, we prove that for any positive integers x and k ≥ 1,
So the inequality (2.19) equivalent to
or
It is clear that the inequality (2.20) holds for all positive integers x and k ≥ 1.
So the inequality (2.19) is true. Using (2.19) repeatedly, we have
Now the inequality (2.15) follows from (2.18) and (2.21).
Similarly, we can consider the case that n = 2m + 1 is odd. Note that xF0(x)F1(x) = 0 and
So Theorem 2 is true if m = 0. If m ≥ 1, then Theorem 2 equivalent to
First we prove that for any positive integers x and k ≥ 1,
So the inequality (2.23) equivalent to
or
It is clear that the inequality (2.24) is correct. So the inequality (2.23) is true.
Using (2.23) repeatedly, we have
On the other hand, we prove that for any positive integers x and k ≥ 1,
So the inequality (2.26) equivalent to
or
It is clear that inequality (2.27) is correct. So the inequality (2.26) is true.
Using (2.26) repeatedly, we have
Now the inequality (2.22) follows from (2.25) and (2.28). This proves Theorem 2.
References
Falcón S, Plaza Á: On the Fibonacci k-numbers, chaos, solitons & fractals. 2007, 32: 1615–1624.
Ma R, Zhang W: Several identities involving the Fibonacci numbers and Lucas numbers. Fibon Q 2007, 45: 164–170.
Yi Y, Zhang W: Some identities involving the Fibonacci polynomials. Fibon Q 2002, 40: 314–318.
Ohtsuka H, Nakamura S: On the sum of reciprocal Fibonacci numbers. Fibon Q 2009, 46/47: 153–159.
Zhang W, Wang T: The infinite sum of reciprocal Pell numbers. Appl Math Comput 2012, 218: 6164–6167. 10.1016/j.amc.2011.11.090
Acknowledgements
The authors express their gratitude to the referee for very helpful and detailed comments. This work is supported by the N.S.F. of P.R.China (11071194).
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Authors' contributions
ZW proposed if we could obtain some similarly results in [4, 5], and proved a part of the theorems. Wu Zhengang obtained the theorems and completed the proof. All authors read and approved the final manuscript.
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Wu, Z., Zhang, W. The sums of the reciprocals of Fibonacci polynomials and Lucas polynomials. J Inequal Appl 2012, 134 (2012). https://doi.org/10.1186/1029-242X-2012-134
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DOI: https://doi.org/10.1186/1029-242X-2012-134