On Chebyshev's Polynomials and Certain Combinatorial Identities

Let T(n)(x) and U(n)(x) be the Chebyshev's polynomial of the first kind and second kind of degree n, respectively. For n >= 1, U(2n-1)(x) = 2Tn(x)U(n-1)(x) and U(2n) (x) = (-1)(n)A(n) (x)A(n)(x), where A(n) (x) = 2(n) Pi(n)(i=1) (x- cos i theta), theta = 2 pi/(2n + 1). In this paper, we will study the polynomial A(n)(x). Let A(n)(x) = Sigma(n)(m-0) a(n,m)x(m). We prove that a(n.m) = (-1)(k)2(m)((l)(k)), where k = left perpendicular n-m/2 right perpendicular and l = left perpendicular n+m/2 right perpendicular. We also completely factorize A(n)(x) into irreducible factors over Z and obtain a condition for determining when A(r)(x) is divisible by A(s)(x). Furthermore we determine the greatest common divisor of A(r)(x) and A(s)(x) and also greatest common divisor of A(r)(x) and the Chebyshev's polynomials. Finally we prove certain combinatorial identities that arise from the polynomial A(n)(x).