On the Sums of Powers of Chebyshev Polynomials and Their Divisible Properties

Lan Zhang,Wenpeng Zhang

doi:10.1155/2017/9363680

Abstract

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.

Highlights

Discussion and ConclusionsLi [1] proved some identities involving power sums of Tn(x) and Un(x)
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 [14], Ozeki [15], Prodinger [16], Melham [17], and Wang and Zhang [18]
For any fixed positive integer h, we prove this polynomial congruence by complete induction for positive integer n

Summary

Discussion and Conclusions

Li [1] proved some identities involving power sums of Tn(x) and Un(x) She obtained some divisibility properties involving Chebyshev polynomials as some applications of these results. We will focus on the problem involving the sums of powers of Chebyshev polynomials These contents are widely used in combinatorial mathematics, and have important theoretical significance for the study of Chebyshev polynomials themselves. 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 [14], Ozeki [15], Prodinger [16], Melham [17], and Wang and Zhang [18].

Several Simple Lemmas

Proof of the Theorem