On Chebyshev's Polynomials and Certain Combinatorial Identities

Abstract

Let Tn(x) and Un(x) be the Chebyshev's polynomial of the rst kind and second kind of degree n, respectively. For n 1, U2n 1(x) = 2Tn(x)Un 1(x) andU2n(x) = ( 1) n An(x)An( x), whereAn(x) = 2 n Q n=1 (x cosi ), = 2= (2n + 1). In this paper, we will study the polynomial An(x). Let An(x) = Pn=0an;mx m . We prove that an;m = ( 1) k 2 m l k , where k = b n m 2 c and l = b n+m 2 c. We also completely factorize An(x) into irre- ducible 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).