An Identity Involving the Integral of the First-Kind Chebyshev Polynomials

Many authors had studied the elementary properties of Chebyshev polynomials and obtained a series of interesting conclusions. For example, C. Cesarano [1], C.-L. Lee and K. B. Wong [2], and Wenpeng Zhang and Tingting Wang [3] proved a series of identities involving Chebyshev polynomials. A. H. Bhrawy et al. (see [4–7]) and N. Bircan and C. Pommerenke [8] obtained many important applications of the Chebyshev polynomials. Xiaoxue Li [9] obtained some identities involving power sums of Tn(x) and Un(x). At the same time, she also proposed the following open problem. Whether there exists 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)? Tingting Wang and Han Zhang [10] partly proved this problem. That is, they proved the identities

The general term formulae of   () and   () are and Many authors had studied the elementary properties of Chebyshev polynomials and obtained a series of interesting conclusions.For example, C. Cesarano [1], C.-L. Lee and K. B. Wong [2], and Wenpeng Zhang and Tingting Wang [3] proved a series of identities involving Chebyshev polynomials.A. H. Bhrawy et al. (see [4][5][6][7]) and N. Bircan and C. Pommerenke [8] obtained many important applications of the Chebyshev polynomials.Xiaoxue Li [9] obtained some identities involving power sums of   () and   ().At the same time, she also proposed the following open problem.
Whether there exists 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)?
They also gave the exact expressions for all constants (, ℎ), (, ℎ), (, ℎ), and (, ℎ) with 1 ≤  ≤  + 1, if  is a small positive integer.If  is large enough, then they only gave an exact computational method for these constants, but the computation is more complex.
For the power sums ∑ ℎ =1 (∫  0   ())  , they have not given any results in [10].Wenpeng Zhang and Tingting Wang [3] and Wenpeng Zhang and Tingting Wang [3] also proposed the following two open problems.Whether there exists a exact calculation formula for In this paper, as a note of [3,9], we shall use the analytic and elementary method to give an interesting computational formula for the second sums of (6).That is, we shall prove the following conclusion.

Several Simple Lemmas
In this section, we shall give several simple lemmas, which are necessary in the proof of our theorem.Hereinafter, we shall use a few basic results, including the properties of sin , for which we refer the reader to the introductory books by Chengdong Pan and Chengbiao Pan [11].First we have the following.
Lemma 2. For any integer  ≥ 0, one has the identity ) . ( Proof.For any real number , from the infinite product of sin(), we have Taking the logarithm for ( 11) and then differentiating it for  we have Let  be a positive integer.Then from the properties of infinite series we have On the other hand, we have ) . ( Taking  → +∞ in ( 14) and then combining ( 12) and ( 13) we may immediately deduce the identity ) . ( This proves Lemma 2.
Lemma 3.For any positive integer , one has the identity Proof.For any positive integer , we have the identity ) . ( Taking  → +∞ in (17), from the method of proving Lemma 2, we have This proves Lemma 3. Proof.In fact this is Lemma 4 of [12].

Proof of the Theorem
Now we shall complete the proof of our theorem.Taking