On the Chebyshev Polynomials and Some of Their Reciprocal Sums

In this paper, we utilize the mathematical induction, the properties of symmetric polynomial sequences and Chebyshev polynomials to study the calculating problems of a certain reciprocal sums of Chebyshev polynomials, and give two interesting identities for them. These formulae not only reveal the close relationship between the trigonometric function and the Riemann ζ-function, but also generalized some existing results. At the same time, an error in an existing reference is corrected.

Since these polynomials have an important position in the theory and application of mathematics, many specialists and scholars have studied their various properties, and obtained a series of interesting conclusions. It is worth mentioning that T. Kim and their team to do a lot of important research work (see [3][4][5][6][7][8]). Other papers related to Chebyshev polynomials can also be found in [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. For example, T. T. Wang and H. Zhang [9] and W. P. Zhang and T. T. Wang [10] obtained some exact expressions for the derivative and integral of the Chebyshev polynomials of the first kind in terms of the Chebyshev polynomials of the first kind. Y. Ma and X. X. Lv [12] considered the calculating problem of a certain reciprocal sums of Chebyshev polynomials, and obtained some identities. That is, for k = 1, 2 and 3, Y. Ma and X. X. Lv [12] gave some identities for the summations where, as usual, q is an odd number and h is an integer co-prime to q, i.e., (h, q) = 1. Unfortunately, it is very difficult to obtain an identity for Equation (2) with k ≥ 4 by the methods in [12]. Inspired by Y. Ma and X. X. Lv [12], in this paper, we utilize the mathematical induction, the properties of symmetric polynomial sequences, and Chebyshev polynomials to study these problems, and prove two generalized conclusions. In other words, we prove the following two results: Theorem 1. Let q be an odd number and q ≥ 3. For any positive integer k and integer h with (h, q) = 1, we have the identity where ζ(s) denotes the Riemann ζ-function, B 2k denotes the Bernoulli numbers, and S(k − 1, i) are defined by Theorem 2. Let q be an odd number and q ≥ 3. For any positive integer k and integer h with (h, q) = 1, we have the identity where we use the identity (see [1], Theorem 12.17) Note that S(0, 0) = S(1, 0) = S(2, 0) = 1, S(1, 1) = 4, S(2, 1) = 20, S(2, 2) = 64, B 2 = 1 6 , B 4 = − 1 30 , and B 6 = 1 42 ; from Theorems 1 and 2, we can immediately deduce the following two corollaries: Corollary 1. [12] Let q > 1 be an odd number. For any integer h and (h, q) = 1, we have the identity

Corollary 2.
[12] Let q > 1 be an odd number. For any integer h and (h, q) = 1, we have the identity Some notes: It is clear that there are some calculation mistakes in [12]. In fact, for k = 3, the corresponding results in [12] are That is to say, Theorems 1 and 2 in [12] are not correct for k = 3. Our theorems obtain a generalized conclusion for all integers k ≥ 1. Thus, our results not only reveal the close connection between a certain trigonometric functions and the Riemann ζ-function, but also generalize some existing results. At the same time, an error in the existing [12] is corrected.
This also reflects the advantages of our theorems. Here, we give partial values of S(k, i), as shown in Table 1.  In Table 1, the first three lines are the values of S(h, i) corresponding to Corollaries 1 and 2, which are no longer listed separately.

Several Lemmas
To facilitate the proofs of our theorems, we need following four basic lemmas.
Proof. From [23] (see Corollary 6, Section 3, Chapter 5), we have the identity Then, from Equation (10) and the properties of the derivative, we also have g(s) = π 2 sin 2 (πs) In general, for any positive integer k, we have Taking s = a q in Equation (12), and then sum over all 1 ≤ a ≤ q − 1. From the definition of the Riemann zeta-function, we have This proves Lemma 3. . For any integer k ≥ 1, we have the identity Proof. If k = 1, then note that (2, q) = 1; from the properties of the complete residue system mod q and the definitions of g(s) and h(s), we have Thus, Lemma 4 is correct for k = 1. Then, note that the identity holds for all positive integers k. Thus, Lemma 4 follows from Equations (13) and (14) and mathematical induction.

Proofs of the Theorems
In this section, we use the lemmas in Section 2 to complete the proofs of our results. First, we prove Theorem 1. For any odd number q ≥ 3 and integer h with (h, q) = 1, taking s = cos πh q , from Lemmas 2 and 3 and the properties of the complete residue system mod q, we have the identity This proves Theorem 1. Now, we prove Theorem 2. Note that the identity f (s) = π 2 cos 2 (πs) = (2π) 2 sin 2 (2πs) − π 2 sin 2 (πs) = h(s) − g(s).
Thus, from this identity and Lemma 1, we have From Equation (15), Lemmas 3 and 4, and the properties of the complete residue system mod q, we have Then, we complete the proof of Theorem 2.

Conclusions
In this paper, we obtain two main results. Theorem 1 establishes a generalized calculation formula for a certain reciprocal sums of Chebyshev polynomials of the first kind. Theorem 2 establishes a generalized calculation formula for a certain reciprocal sums of Chebyshev polynomials of the second kind. As two special cases or two corollaries of our theorems, we give a new proof of the results in [12], and we also point out two computational errors in [12].