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)An(x)An(−x), whereAn(x) = 2n ∏n i=1(x− cos iθ), θ = 2π/(2n + 1). In this paper, we will study the polynomial An(x). Let An(x) = ∑n m=0 an,mx m. We prove that an,m = (−1)k2m ( l k ) , where k = bn−m 2 c and l = b 2 c. 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 Chebyshev’s polynomials. Finally we prove certain combinatorial identities that arise from the polynomial An(x). 2010 Mathematics Subject Classification: 11R09, 13A05, 05A19