On Chebyshev Polynomials, Fibonacci Polynomials, and Their Derivatives

. We study the relationship of the Chebyshev polynomials, Fibonacci polynomials


Introduction
As we know, the Chebyshev polynomials and Fibonacci polynomials are usually defined as follows: the first kind of Chebyshev polynomials is +2 ( ) = 2 +1 ( ) − ( ) and ≥ 0, with the initial values 0 ( ) = 1 and 1 ( ) = ; the second kind of Chebyshev polynomials is +2 ( ) = 2 +1 ( )− ( ) and ≥ 0, with the initial values 0 ( ) = 1 and 1 ( ) = 2 ; the Fibonacci polynomials are +2 ( ) = (1) 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 +2 = +1 + , where ≥ 0, 0 = 0, 1 = 1, 0 = 2, and 1 = 1. 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 ( ) 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][5][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 −1.

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

Lemma 7.
For any positive integers and , one has the following identities: Proof. See [3].
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.
where ( ) ( ) denotes the th derivative of ( ) with respect to . Then one can get Proof. To begin with, we multiply √ 1 − 2 ( ) to both sides of the following identity: and then integrate it from −1 to 1. Applying property (10), we can get and then we have We define where and are any nonnegative integers. Let = cos ; then we can get the following identity by applying property (10): According to Lemma 9 and property (27), we have Then we have 2 , , = 0 if − is odd. If − is even, we have This proves property (21). In the similar way, we have That is property (22). This proves Lemma 10. (32) Then one can get Proof. In order to prove property (22) we must multiply ( )/ √ 1 − 2 to both sides of the following identity: and then integrate it from −1 to 1. Applying property (9) we can get and then we have We define From [8], we know where and are any nonnegative integers. Let = cos ; then we can get the following identity by applying property (10): This proves property (33). In the similar way 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, (46) Let = 2 cos ; then we have This proves property (44). Let = 2 cos in the following identity: Proof. At first, we multiply √ 2 + 4 ( ) to both sides of the following identity: and then integrate it from −2 to 2 ; we can get the following identity by applying Lemma 12, where is any positive integer. Consider Let = cos ; then we can get the following identity by applying Lemma 12: According to property (27), we have In the similar way, we have 2 +1, , = 0 if − is odd. If − is even, we can get This proves Lemma 13.

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 = /2 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 = /2 in the above identities, according to Lemma 7, we can get Corollary 5.