Complex Factorization by Chebysev Polynomials

Abstract

Let $\left \{ a_{i}\right \}, \left \{ b_{i}\right \}$ be real numbers
for $0\leqslant i\leqslant r-1$, and define a \textit{$r$-periodic sequence}
$\left\{v_{n}\right \}$ with initial conditions $v_{0}$, $v_{1}$ and recurrences
$v_{n}=a_{t}v_{n-1}+b_{t}v_{n-2}$ where $n \equiv t \left( \text{mod } r\right)$ ($n\geqslant 2$).
In this paper, by aid of Chebyshev polynomials, we introduce a new method to obtain the complex factorization of
the sequence $\left\{v_{n}\right \}$ so that we extend some recent
results and solve some open problems. Also, we provide new results by obtaining
the binomial sum for the sequence $\left\{v_{n}\right \}$ by using Chebyshev polynomials.