On the Chebyshev polynomials and some of their new identities

The main purpose of this paper is, using the elementary methods and properties of the power series, to study the computational problem of the convolution sums of Chebyshev polynomials and Fibonacci polynomials and to give some new and interesting identities for them.

We all know that the polynomials T n (x) and U n (x) play important roles in the study of orthogonality of functions and approximation theory, so many scholars have studied their properties and obtained a series of valuable research results. In particular, in the references we have seen that Kim and his team have done a lot of important research work (see [3][4][5][6][7][8][9][10][11]), and Cesarano (see [12][13][14]) has also made a lot of contributions. Some other papers related to these polynomials and sequences can be found in references [2,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. For example, Zhang Wenpeng [17] studied the calculating problem of the convolution sums of T n (x) and proved the following: where a 1 +a 2 +···+a k+1 =n denotes that the summation is taken over all (k + 1)-dimension nonnegative integer coordinates (a 1 , a 2 , . . . , a k+1 ) such that a 1 + a 2 + · · · + a k + a k+1 = n, U (k) n (x) denotes the kth derivative of U n (x) with respect to x. Zhang Yixue and Chen Zhuoyu [18] proved the following result: where C(h, i) is a second-order nonlinear recurrence sequence defined by C(h, 0) = 0, C(h, h) = 1, C(h + 1, 1) = 1 · 3 · 5 · · · (2h -1) = (2h -1)!!, and C(h + 1, i + 1) = (2h -1 - Obviously the results in [17,18], and [21] do not look very concise and clear. It is even harder to calculate their exact values. Inspired by these papers, we also became interested in such problems and used different methods to come up with simpler, more beautiful identities. That is, we use the elementary methods and the properties of the power series to prove the following conclusions: Theorem 1 For any integers k ≥ 2 and n ≥ 0, we have the identity Theorem 2 For any integer k ≥ 2 and integer n ≥ 0, we have For Lucas polynomials L n+2 (x) = xL n+1 (x) + L n (x) with L 0 (x) = 2, L 1 (x) = x and Fibonacci polynomials F n+2 = xF n+1 + F n (x) with F 0 (x) = 0 and F 1 (x) = 1, we can also deduce the following corresponding results: Theorem 3 For any positive integer k ≥ 2 and integer n ≥ 0, we have Theorem 4 For any positive integer k ≥ 2 and integer n ≥ 0, we have , it is clear that our Theorem 4 is much simpler than the corresponding identity in Ma Yuankui and Zhang Wenpeng [20]. Taking k = 3, 4 or k = 5 with x = 1, from Theorems 2 and 4 we can deduce the following four corollaries: Corollary 2 For any integer n ≥ 0, we have the identity Corollary 3 For any integer n ≥ 0, we have the identity (-1) i (i + 1)(i + 2)(i + 3)(ni + 1)(ni + 2)(ni + 3)L n-2i (x).

Corollary 4
For any integer n ≥ 0, we have the identity For any nonnegative integers m and n, note that T m (T n (x)) = T mn (x). From Theorem 1 we can also deduce the following: Corollary 5 For any integers k ≥ 2 and m ≥ 0, we have the identity a 1 +a 2 +···+a k =n Some notes: It is worth noting that Theorem 1 has been obtained by different methods in equation (29) of [9], but the expression is different from our result. In fact equation (29) in [9] involved the Gauss hypergeometric function, so it looks a little bit more complicated, and our Theorem 1 is simple and straightforward. Theorem 2 has been obtained by different methods in (1.30) of [3].

Proofs of the theorems
Now we prove our main results directly. First, we will prove Theorem 1. From the generating function (2) of T n (x), we have 1 2 So, for any positive integer k ≥ 2, from (3) and the properties of the power series, we have the identity On the other hand, note the power series For any positive integers r and h, we have where α · β = 1. From (6) and the definition of T n (x), we have From (5), (7), the definition and properties of the binomial, we have Combining (4) and (8) and then comparing the coefficients of the power series, we have the identity a 1 +a 2 +a 3 +···+a k =n T a 1 (x) · T a 2 (x) · T a 3 (x) · · · T a k (x) This proves Theorem 1.
To prove Theorem 2, note that, for any positive integer k, From (5) and (9) we have ∞ n=0 a 1 +a 2 +···+a k =n Comparing the coefficients of t n in (10), we have the identity This proves Theorem 2.
, then from the definition of L n (x) we have L n (x) = γ n + δ n and Note that γ · δ = -1, from (5), (6), and the methods of proving Theorem 1, we have ∞ n=0 a 1 +a 2 +···+a k =n Comparing the coefficients of t n in (12), we have This proves Theorem 3. Similarly, from the methods of proving Theorem 2, we can also deduce a 1 +a 2 +···+a k =n This completes the proofs of all our theorems.

Conclusion
The main results of this paper are four theorems and five corollaries. Theorem 1 established an identity for the convolution sums of Chebyshev polynomials of the first kind. This improved an early result in [17] and [21]. Theorem 2 simplified the identity in [18] and made it look more concise and beautiful. It must be noted that Theorem 1 and Theorem 2 appear in different forms in other references, such as [3] and [9]. From Theorem 3 and Theorem 4, we can get two corresponding results for Fibonacci polynomials and Lucas polynomials. In addition, in Theorem 4 we have improved a new result in [20]. The five corollaries are just some special cases of our four theorems. These results are actually new contributions to the study of the properties of Chebyshev polynomials and Fibonacci polynomials. Of course, the methods adopted in this paper have some good reference for further study of the properties of general second-order linear recursive sequences.