On Chebyshev's Polynomials and Certain Combinatorial Identities

Chan-Lye Lee ; K.B. Wong

Bulletin of the Malaysian Mathematical Society, Tome 34 (2011) no. 2

Résumé

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 $A_{n}(x)$.

Détail

Classification : 11R09, 13A05, 05A19.