On Chebyshev s Polynomials and Certain Combinatorial Identities

Let Tn(x) and Un(x) be the Chebyshev s polynomial of the first kind and second kind of degree n, respectively. For n ≥ 1, U2n-1(x) = 2Tn(x)Un-1(x) and U2n(x) = (-1)nAn(x)An(-x), where An(x) = 2^n ∏^n i =1^(x- cos iθ), θ = 2(pi)/(2n + 1). this paper, we will study the polynomial An(x). Let An(x) = Σ^n m =0^an,m^x^m. We prove that an,m = (-1)^k 2^m (left( begin{array}{c} l k end{array} right), where k = [n-m/2] and l = [n+m/2]. We also completely factorize An(x) into irreducible factors over Z and obtain a condition for determining when Ar(x) is divisible by As(x). Furthermore we determine the greatest common divisor of Ar(x) and As(x) and also greatest common divisor of Ar(x) and the Cheby-shev s polynomials. Finally we prove certain combinatorial identities that arise from the polynomial An(x).