Abstract
We use the elementary and analytic methods and the properties of Chebyshev polynomials to study the computational problem of the reciprocal sums of one-kind Chebyshev polynomials and give several interesting identities for them. At the same time, we also give a general computational method for this kind of reciprocal sums.
1. Introduction
It is well known that Chebyshev polynomials of the first and second kind and are defined as follows: , , and the recursion formula for all integers . , , and the recursion formula for all integers . The generation functions of these polynomials are The general term formulae of and are expressed asIf we take , then
Since all these definitions and properties of Chebyshev polynomials can be found in any handbook of mathematics, there is no need to list the source everywhere.
Recently, some authors studied the properties of Chebyshev polynomials and obtained many interesting conclusions. For example, Li [1] obtained some identities involving power sums of and . As some applications of these results, she obtained some divisibility properties involving Chebyshev polynomials. 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) is the question.
Wang and Zhang [2] and Zhang and Wang [3] partly solved these problems. Some theoretical results related to Chebyshev polynomials can be found in Ma and Zhang [4], Cesarano [5], Babusci et al. [6–8], Lee and Wong [9], and Wang and Han [10]. Doha and others [11–14] and Bircan and Pommerenke [15] also obtained many important applications of the Chebyshev polynomials.
In this paper, we consider the computational problem of the reciprocal sums of Chebyshev polynomials. That is, let and be positive integers with , for any real number ; if , then we consider the computational problem of the summations
Although there are many results related to Chebyshev polynomials, it seems that none had studied the computational problem of (4). The main reason may be that a computational formula does not exist. But for some special real number , we can really get the precise value of (4). In this paper, we will illustrate this point. That is, we will use the elementary and analytic methods and the properties of Chebyshev polynomials to prove the following results.
Theorem 1. Let be an integer with . Then for any integer with , one has the identities
Theorem 2. Let be an odd number with . Then for any integer with , one has the identities
Some Notes. First in Theorem 2, we must limit as an odd number. Otherwise, if is an even number, then is an integer, and . Therefore, the fraction is meaningless.
Second, for any positive integer and with , we can give an computational formula for (4). Of course, the calculation is very complicated when is larger. So we do not give a general conclusion for (4), only give an efficient calculating method. In fact if we use computer MatLab program, and by means of recursive method in Lemma 4, we can also obtain all precise values of (4) for any positive integer .
2. Several Lemmas
To complete the proofs of our theorems, we need following lemmas. First we have
Lemma 3. Let be an integer. Then for variable with and function , one has the identity where denotes the -order derivative of , is Bernoulli numbers.
Proof. For any real number , from Corollary (Section , Chapter ) in [16] we have the identityThen from (8), the definition and properties of derivative we have Generally, for any positive integer , we haveNow taking in (10), and summation for all and noting the definition and properties of complete residue system (see [16]) we haveNote that Riemann -function (see [17], Theorem 12.17). Then from (11) we have This proves Lemma 3.
Lemma 4. Let , . Then one has
Proof. Noting that the identity , from the definition and properties of derivative we have and This proves the first formula of Lemma 4.
Similarly, we have This is the second formula of Lemma 4.
It is easy to prove that This completes the proof of Lemma 4.
Lemma 5. Let be an integer with . Then for any integer with , one has the identities
Proof. Since , if pass through a complete residue system , then also pass through a complete residue system . Therefore, without loss of generality we can assume that . Noting that and , from Lemmas 3 and 4 with we have orSimilarly, noting that , from Lemmas 3 and 4 with and applying (19) we have which implies that orNoting that , from Lemmas 3 and 4 with , applying (19) and (22) we have or which implies thatNow Lemma 5 follows from (19), (22), and (25).
In fact, by using Lemma 4 and the method of proving Lemma 5 we can give a computational formula for with all positive integer . Here just in order to meet the demands of main results we only calculated , and .
3. Proofs of the Theorems
In this section, we shall complete the proofs of our theorems. First we prove Theorem 1. For any integer , taking with , from (3) we haveNow Theorem 1 follows from (27) and Lemma 5 with , and .
To prove Theorem 2, we note that, for any odd number , if pass through a complete residue system , then also pass through a complete residue system . So from the properties of trigonometric functions we haveFrom (3), (28), and Lemma 5 we may immediately deduce that
Similarly, we also haveSo from (30) and Lemma 5 we haveCombining (3) and (31) we have the identityFrom the method of proving (30) we also haveFrom (3), Lemma 5, and (33) we can deduce thatNow Theorem 2 follows from (29), (32), and (34).
This completes all proofs of our results.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by NSF (11371291) of China and the Scientific Research Program Funded by Shaanxi Provincial Education Department (no. 16JK1373).