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

Abstract

For any integer ≥ 0, the first-kind Chebyshev polynomials { ( )} and the second-kind Chebyshev polynomials { ( )} are defined as 0 ( ) = 1, 1 ( ) = , 0 ( ) = 1, 1 ( ) = 2 and +2 ( ) = 2 +1 ( ) − ( ), +2 ( ) = 2 +1 ( ) − ( ) for all ≥ 0. If we write = + √ 2 − 1 and = − √ 2 − 1, then we have 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.