Research Article | Open Access
On the Sums of Powers of Chebyshev Polynomials and Their Divisible Properties
The main purpose of this paper is using mathematical induction and the Girard and Waring formula to study a problem involving the sums of powers of the Chebyshev polynomials and prove some divisible properties. We obtained two interesting congruence results involving Fibonacci numbers and Lucas numbers as some applications of our theorem.
1. Discussion and Conclusions
For any integer , the first-kind Chebyshev polynomials and the second-kind Chebyshev polynomials are defined as , , , and , for all . If we write and , then we have Since Chebyshev polynomials occupy very important position in the theory and application of mathematics, many scholars have studied their various properties and obtained a series of important results. See [1–11]. For example, Li  proved some identities involving power sums of and . She obtained some divisibility properties involving Chebyshev polynomials as some applications of these results. Ma and Lv  studied the computational problem of the reciprocal sums of Chebyshev polynomials and obtained some identities. Some theoretical results related to Chebyshev polynomials can be found in Ma and Zhang , Cesarano , Lee and Wong , Bhrawy and others (see [6–9]), and Wang and Zhang . Bircan and Pommerenke  also obtained many important applications of the Chebyshev polynomials.
In this paper, we will focus on the problem involving the sums of powers of Chebyshev polynomials. These contents not only are widely used in combinatorial mathematics, but also have important theoretical significance for the study of Chebyshev polynomials themselves. Here we will use mathematical induction and the Girard and Waring formula (see [12, 13]) to prove some interesting divisible properties for Chebyshev polynomials. That is, we will prove the following.
Theorem 1. Let and be any positive integers; then we have the congruence Taking and in and , then we have where and are Fibonacci numbers and Lucas numbers, respectively.
There are many very interesting and important results related to Fibonacci numbers and Lucas numbers; some of them can be found in Yi and Zhang , Ozeki , Prodinger , Melham , and Wang and Zhang .
We can also deduce interesting congruence properties involving Fibonacci numbers and Lucas numbers as an application of our theorem. That is, we have the following corollaries.
Corollary 2. For any positive integers and , we have the congruence
Corollary 3. For any positive integers and , we have the congruence
2. Several Simple Lemmas
To complete the proof of our main result, we need several simple lemmas. First we have the following.
Lemma 4. For any positive integer , we have
Proof. If , then and . It is clear that and So without loss of generality we can assume that . Then from the definition of and binomial theorem we have the identity From (8), we have the polynomial congruence or which implies that On the other hand, from (8) we also have which implies thatIt is clear that Lemma 4 follows from (11) and (13).
Lemma 5. Let and be any positive integers; then we have the polynomial congruence
Proof. For any fixed positive integer , we prove this polynomial congruence by complete induction for positive integer . Note that , , , , and , so if , then we have That is to say, Lemma 5 is true for .
If , then note that the identities we have the polynomial congruence So Lemma 5 is true for .
Suppose Lemma 5 is true for all integers . That is, for all .
Then, for , note that the identities from inductive assumption (18), we have Now Lemma 5 follows from complete induction.
Lemma 6. For any integers and , we have the congruence
Proof. For integer , note that the identities , , It is clear that from (22) we know that, to prove Lemma 6, we need only to prove the polynomial congruence Now we use complete induction to prove (23). It is clear that polynomial congruence (23) is true for . If , then note that , , and ; we have So congruence (23) is true for . Suppose (23) is true for all integers . Namely, for all .
Then, for , note that the identities from inductive assumption (25) and Lemma 5, we have This proves Lemma 6 by complete induction.
Lemma 7. For all nonnegative integers and real numbers and , we have the identity where denotes the greatest integer .
3. Proof of the Theorem
In this section, we will prove our theorem by mathematical induction. Taking , , and in Lemma 7, note that ; from the definition of and binomial theorem, we have For any integer , from (29) we have the identity Taking in (30), we have the identity From Lemma 4, we know that , so applying Lemma 6 and (31) we can deduce that That is to say, the theorem is true for .
Suppose that the theorem is true for all integers . That is, for all integers .
Then, for , from (29) we have From Lemma 6, we have Applying inductive assumption (33), we have Combining (34), (35), (36), and Lemma 6, we can deduce the polynomial congruence From Lemma 4, we know that , so from (37) we may immediately deduce the polynomial congruence This completes the proof of our theorem by mathematical induction.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is supported by NSF China (Grant no. 11771351).
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