Abstract
The main purpose of this paper is using the combinatorial method and algebraic manipulations to study some sums of powers of Chebyshev polynomials and give several interesting identities. As some applications of these results, we obtained several divisibility properties involving Chebyshev polynomials.
1. Introduction
For any integer , the famous Chebyshev polynomials of the first and second kind and are defined as follows: , , and for all ; , , and for all .
It is clear that these polynomials are the second-order linear recurrence polynomial; they satisfy the computational formulae:
About the elementary properties of Chebyshev polynomials and related second-order linear recurrences, many authors had studied them and obtained a series of interesting conclusions. For example, some of the theoretical results can be found in [1–4], and other some important applications of the Chebyshev polynomials can also be found in [5–10].
Recently, several authors studied the sums of powers of Fibonacci numbers and Lucas numbers , and obtained a series of important identities; see [11–13]. At the same time, Melham [13] also proposed the following two conjectures.
Conjecture 1. Let be an integer. Then the sum can be expressed as , where is a polynomial of degree with integer coefficients.
Conjecture 2. Let be an integer. Then the sum can be expressed as , where is a polynomial of degree with integer coefficients.
Wang and Zhang [14] solved the Conjecture 2 completely and made some substantial progress for the Conjecture 1.
The main purpose of this paper is using the algebraic manipulations to obtain some identities involving Chebyshev polynomials of the first and second kind and . As some applications, we give three interesting corollaries. That is, we will prove the following two results.
Theorem 3. For any positive integers and , we have the identities (a)(b)
Theorem 4. For any positive integers and , we have the identities (A)(B)
The benefit of these identities is that it can transform the complex sums of powers of Chebyshev polynomials that become relatively simple linear sums of Chebyshev polynomials. This can simplify the calculation problems related to the sums of powers of Chebyshev polynomials.
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 an open problem. we will be looking for some new methods to further research.
Note that and ; it is clear that from Theorems 3 and 4 we can deduce some identities involving and . On the other hand, we can also obtain some divisibility properties involving Chebyshev polynomials. That is, we have the following.
Corollary 5. Let and be two integers. Then the sum can be divided by polynomials .
The sum can be divided by polynomials .
Corollary 6. Let and be two integers. Then the sum can be expressed as , where is an integer coefficients polynomial of two variables with degree of .
Corollary 7. Let and be two integers. Then the sum can be expressed as , where is an integer coefficients polynomial of two variables with degree of .
2. Proof of the Theorems
In this section, we will use the algebraic manipulations to complete the proof of our theorems. First we prove Theorem 3. In fact, for any positive integer and real number , by using the familiar binomial expansion we may get
Now we take in (13); then note that ; from the definitions of and , we may immediately deduce the identitiesIf we take in (13), then we can also deduce the identitiesLet and . Then for any integer , note that ; from (14) and the definitions of and we have This proves the identity (a) of Theorem 3.
Similarly, from formula (15) we can deduce the identity (b) of Theorem 3.
Now we prove Theorem 4. From (16) we have This proves the identity (A) of Theorem 4.
Similarly, from formula (17) we can also deduce the identity (B) of Theorem 4.
Now we use Theorem 3 to prove Corollary 5. From the properties of Chebyshev polynomials we know that and ; from (a) of Theorem 3 we may immediately deduce that That is, the power sum can be divided by polynomial .
Similarly, note that ; from (b) of Theorem 3 we know that divide the power sum This proves Corollary 5.
Now we use (B) of Theorem 4 to prove Corollary 7. It is clear that if is an integer coefficients polynomial with variable , then, for any polynomials and , we have divides . From those properties and noting that the identity we can deduce thatFrom (23) and (B) of Theorem 4 we can deduce that where is an integer coefficients polynomial of two variables with degree of . This proves Corollary 7.
Finally, we prove Corollary 6. We first prove thatfor all integers . It is clear that if , then the conclusion is correct. So without loss of generality we can assume . Note that the identity , so we have Therefore, to prove (25), we only need to prove thatSince is an even function, from we can deduce that ; note that the coprime relations , so from the properties of polynomials we know thatSince , from (28) we know that to prove (27), we only need to proveNote that the identity From this identity we may immediately deduce (29). That is, (25) is correct.
Applying (25) and (A) of Theorem 4 we can easily deduce that the sum can be expressed as , where is an integer coefficients polynomial of two variables with degree of .
This completes the proofs of our all results.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the P. S. F. (2014JM1009) and N. S. F. (11371291).