Journal of Applied Mathematics

Volume 2014 (2014), Article ID 451953, 8 pages

http://dx.doi.org/10.1155/2014/451953

Research Article

On Chebyshev Polynomials, Fibonacci Polynomials, and Their Derivatives

Yang Li

Department of Mathematics, Northwest University, Xi’an, Shaanxi 710127, China

Received 6 June 2013; Revised 19 November 2013; Accepted 7 February 2014; Published 13 March 2014

Academic Editor: Vijay Gupta

Copyright © 2014 Yang Li. 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 study the relationship of the Chebyshev polynomials, Fibonacci polynomials, and their rth derivatives. We get the formulas for the rth derivatives of Chebyshev polynomials being represented by Chebyshev polynomials and Fibonacci polynomials. At last, we get several identities about the Fibonacci numbers and Lucas numbers.

1. Introduction

As we know, the Chebyshev polynomials and Fibonacci polynomials are usually defined as follows: the first kind of Chebyshev polynomials is and , with the initial values and ; the second kind of Chebyshev polynomials is and , with the initial values and ; the Fibonacci polynomials are and with the initial values and . From the second-order linear recurrence sequences, we have

These polynomials play a very important role in the study of the theory and application of mathematics and they are closely related to the famous Fibonacci numbers and Lucas numbers which are defined by the second-order linear recurrence sequences

where , , , , and . Therefore, many authors have investigated these polynomials and got many properties and corollaries. For example, Wu and Zhang [1] have obtained the general formulas involving

where is any positive integer. Wu and Yang [2] studied Chebyshev polynomials and got a lot of properties.

Recently, several authors also studied the derivatives of these polynomials. For example, Zhang [3] used the th derivatives of Chebyshev polynomials to solve some calculating problems of the general summations. Falcón and Plaza [4–6] presented many formulas and relations between Fibonacci polynomials and their derivatives. This fact allows them to present a family of integer sequences in a new and direct way.

In this paper, we combine Sergio Falcón and Wenpeng Zhang's ideas. Then we obtain the following theorems and corollaries. These results strengthen the connections of two kinds of polynomials. They are also helpful in dealing with some calculating problems of the general summations or studying some integer sequences.

Theorem 1. For any positive integers, and , one has the following formulas:

where denotes the th derivative of with respect to .

Theorem 2. For any positive integers, and , one has the following formulas:

Theorem 3. For any positive integers and one has the following formulas:

Corollary 4. For any positive integers , , and , one has the following identities:

where denotes the square root of .

Corollary 5. For any positive integers , , and , one has the following identities:

2. Some Lemmas

Lemma 6. For any nonnegative integers and , one has the following identities:

Proof. See [7].

Lemma 7. For any positive integers and , one has the following identities:

Proof. See [3].

Lemma 8. For any positive integer , one has

Proof. See [3].

Lemma 9. For any positive integers and , one has

Proof. From Theorem 2 of [2], we can get the following result easily:

From Theorem 2 of [2], we know

In the similar way, we can get the following result easily:

If we derive both sides of the above properties th times, we will get

This proves Lemma 9.

Lemma 10. For any positive integers and , let

where denotes the th derivative of with respect to . Then one can get

Proof. To begin with, we multiply to both sides of the following identity:

and then integrate it from to . Applying property (10), we can get

and then we have

We define

From [8], we know

where and are any nonnegative integers. Let ; then we can get the following identity by applying property (10):

According to Lemma 9 and property (27), we have

Then we have if is odd. If is even, we have

This proves property (21). In the similar way, we have if is even. If is odd, we have

That is property (22). This proves Lemma 10.

Lemma 11. For any positive integers and , let

Then one can get

Proof. In order to prove property (22) we must multiply to both sides of the following identity:

and then integrate it from to . Applying property (9) we can get

and then we have

We define

From [8], we know

where and are any nonnegative integers. Let ; then we can get the following identity by applying property (10):

According to Lemma 9 and property (39), we have

so we have if is odd. If is even, we have

This proves property (33). In the similar way we have if is even. If is odd, we have

That is property (34). This proves Lemma 11.

Lemma 12. For any positive integers and , one has the following identities:

Proof. As we know,

Let ; then we have

This proves property (44). Let in the following identity:

then we can get

Then we can get property (45). This proves Lemma 12.

Lemma 13. For any positive integer , let

then we can get

Proof. At first, we multiply to both sides of the following identity:

and then integrate it from to ; we can get the following identity by applying Lemma 12, where is any positive integer. Consider

then we have

Let ; then we can get the following identity by applying Lemma 12:

According to property (27), we have

so we have if is even. If is odd, we can get

In the similar way, we have if is odd. If is even, we can get

This proves Lemma 13.

3. Proof of the Theorems and Corollaries

In this section, we will prove our theorems and corollaries. First of all, we can prove all the theorems from Lemmas 10, 11, and 13 easily. Then we prove our corollaries.

Proof of Corollary 4. Let in Theorem 1. We can get the following properties from Lemma 8:

Then, taking in the above identities, according to Lemma 7, we can get Corollary 4.

Proof of Corollary 5. Let in Theorem 2. We can get the following properties from Lemma 8:

Then, taking in the above identities, according to Lemma 7, we can get Corollary 5.

Conflict of Interests

The author declares that he has no conflict of interests in this paper.

Acknowledgments

The author would like to thank the referee for his very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the N.S.F. (11371291, 61202437) and S.R.F.D.P. (20136101110014) of China.

References

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