- Research
- Open Access
On Chebyshev polynomials and their applications
- Xingxing Lv1 and
- Shimeng Shen1Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-017-1387-8
© The Author(s) 2017
- Received: 7 September 2017
- Accepted: 2 October 2017
- Published: 24 October 2017
Abstract
The main purpose of this paper is, using some properties of the Chebyshev polynomials, to study the power sum problems for the sinx and cosx functions and to obtain some interesting computational formulas.
Keywords
- Chebyshev polynomials
- trigonometric power sums
- computational formulas
MSC
- 11B39
1 Introduction
As far as we know, it seems that nobody has studied these problems yet. In this paper, we shall use the properties of the Chebyshev polynomials of the first kind to obtain some closed formulas for the above trigonometric power sums. These results are stated in the following theorems.
Theorem 1
Theorem 2
Theorem 3
It is clear that from Theorem 1 and Theorem 2 we may immediately deduce the following corollary.
Corollary
Remark 1
2 Several simple lemmas
To complete the proofs of our theorems, we need some new properties of Chebyshev polynomials, which we summarize as the following lemmas.
Lemma 1
Proof
Lemma 2
Proof
This is Lemma 4 in Ma Rong and Zhang Wenpeng [7]. □
Lemma 3
Proof
3 Proofs of the theorems
In this section, we shall complete the proofs of our theorems. First we prove Theorem 1.
Proof of Theorem 1
Now we prove Theorem 2.
Proof of Theorem 2
We now use the similar methods of proving Theorem 1 and Theorem 2 to complete the proof of Theorem 3.
Proof of Theorem 3
Note that if \(2\mid q\), then \(q\nmid(2k+1)\) for any non-negative integer k. So, by substituting \(\sin(\frac{\pi a}{q})\) for x in the second formula of Lemma 2 and making the summation for a with \(0\leq a \leq q-1\), with the help of Lemma 1 and Lemma 3, we deduce Theorem 3 immediately. □
Declarations
Acknowledgements
The authors would like to thank the referees for their very helpful and detailed comments which have significantly improved the presentation of this paper. This work was supported by the N.S.F. (Grant No. 11771351) of P.R. China.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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