Some Identities Involving the Derivative of the First Kind Chebyshev Polynomials

About the other various properties of the Chebyshev polynomials, some authors had studied them and obtained many interesting conclusions. For example, Ma and Zhang [1], Wang and Han [2], Cesarano [3], and Lee and Wong [4] proved a series of identities involving Chebyshev polynomials. Bhrawy and others (see [5–10]) obtainedmany important applications of the Chebyshev polynomials. Very recently, Li [11] proved some identities involving power sums of T n (x) and U n (x). That is, for any positive integers h and n, one has the identities:


Introduction
For any integer ≥ 0, the famous Chebyshev polynomials of the first and second kind ( ) and ( ) are defined as follows: where [ /2] denotes the greatest integer ≤ /2. It is clear that ( ) and ( ) are the second-order linear recurrence polynomials; they satisfy the recurrence formulae: Both ( ) and ( ) are orthogonal polynomials. That is, , if = > 0; , if = = 0, , if = . ( About the other various properties of the Chebyshev polynomials, some authors had studied them and obtained many interesting conclusions. For example, Ma and Zhang [1], Wang and Han [2], Cesarano [3], and Lee and Wong [4] proved a series of identities involving Chebyshev polynomials. Bhrawy and others (see [5][6][7][8][9][10]) obtained many important applications of the Chebyshev polynomials.
Very recently, Li [11] proved some identities involving power sums of ( ) and ( ). That is, for any positive integers ℎ and , one has the identities: Mathematical Problems in Engineering As for some applications of these results, Xiaoxue Li obtained some divisibility properties involving Chebyshev polynomials. At the same time, she also proposed the following open problem.
Does there exist an exact expression for the derivative or integral of the Chebyshev polynomials of the first kind in terms of the Chebyshev polynomials of the first kind (and vice versa)? That is to say, does there exist an exact expression for the summations in terms of the Chebyshev polynomials of the first kind?
Does there exist an exact expression for ∑ ℎ =1 ( ) in terms of ( ) or ∫ 0 ( ) where ( ) denotes the derivative of ( ) with respect to ?
In this paper, as a note of [11], we give some identities involving the derivative of the first kind Chebyshev polynomials. That is, we will prove the following. Theorem 1. For any integers ℎ ≥ 1 and ≥ 0, one has the identity where ( , ℎ), ( , ℎ), ( , ℎ), and ( , ℎ) are computable constants.
For some special , from Theorem 1 with = 0 and Theorem 2 with = 1, we can also deduce the following two corollaries.

Corollary 3. For any positive integer ℎ, one has the identity
Corollary 4. For any positive integer ℎ, one has the identity (12)

Two Simple Lemmas
In this section, we will give two simple lemmas, which are necessary in the proofs of our theorems. First we express ( ( )) 2 and ( ( )) 2 +1 in terms of ( ) and ( ). That is, we have the following.

Lemma 1. For any positive integers and , one has the identities
Proof. For any positive integer and real number ̸ = 0 and 1, by using the familiar binomial expansion we have Now we take = ( + √ 2 − 1) in (14); note that 1/ = ( − √ 2 − 1) ; from the definition of ( ) and ( ) we have the identities Note that ( ) = −1 ( ); from (15) we may immediately deduce This proves Lemma 1.

Lemma 2.
Let ℎ and be two positive integers. Then for any ̸ = 0 and 1, we have the recurrence formula: There exist some computable constants ( , ) and ( , ) such that In particular for = 1 and 2, we have the identities: ; it is clear that Note that the binomial expansion is as follows: and from (21) we may immediately deduce the identity: This proves the recurrence formula (I).
It is easy to prove (II) by recurrence formula (I) and the complete mathematical induction.
From the recurrence formula (I) and noting that the identity we can also deduce This proves formula (A). Identity (B) follows from (24), (A), and the recurrence formula (I) with = 2. This proves Lemma 2.
Some Note. The first part of Lemma 2 obtained an interesting recurrence formula for the computation of the summation ∑ ℎ

=1
. But if positive integer is large enough, then the computation of the recurrence formula is more complex, and so we have not given the exact constants ( , ℎ) and ( , ℎ) in formula (II).
Some Comments. In our theorems, we can give the exact expressions for all constants ( , ℎ), ( , ℎ), ( , ℎ), and ( , ℎ), if is a small positive integer. If is large enough, then we can only give an exact computational method for these constants (because of the reason of Lemma 2), but the computation is more complex, and so we have not obtained the exact expression for ( , ℎ), ( , ℎ), ( , ℎ), and ( , ℎ), 1 ≤ ≤ ℎ.