On Chebyshev Polynomials, Fibonacci Polynomials, and Their Derivatives

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.