Volume 34 • Number 2 • 2011   • On Chebyshev's Polynomials and Certain Combinatorial Identities Chan-Lye Lee and K.B. Wong Abstract. 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 \(n\geq 1\), \(U_{2n-1}(x)=2T_{n}(x)U_{n-1}(x)\) and \(U_{2n}(x)=(-1)^nA_n(x)A_{n}(-x)\), where \(A_n(x)=2^n\prod_{i=1}^n (x-\cos i\theta)\), \(\theta=2\pi/(2n+1)\). In this paper, we will study the polynomial \(A_n(x)\). Let \(A_n(x)=\sum_{m=0}^n a_{n,m} x^m\). We prove that \(a_{n,m}=(-1)^k2^m {l \choose k}\), where \(k=\lfloor \frac{n-m}{2}\rfloor\) and \(l=\lfloor \frac{n+m}{2}\rfloor\). We also completely factorize \(A_n(x)\) into irreducible factors over \(\mathbb Z\) 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 \(A_{n}(x)\). 2010 Mathematics Subject Classification: 11R09, 13A05, 05A19. Full text: PDF