grandes-ecoles 2014 QII.A.1

grandes-ecoles · France · centrale-maths2__mp Roots of polynomials Proof of polynomial identity or inequality involving roots

The Chebyshev polynomials of the first and second kind are defined by $T_n(\cos\theta) = \cos(n\theta)$ and $U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$ for $\theta \notin \pi\mathbb{Z}$.
Show that $$\begin{cases} T_m \cdot T_n = \frac{1}{2}\left(T_{n+m} + T_{n-m}\right) & \text{for all integers } 0 \leqslant m \leqslant n \\ T_m \cdot U_{n-1} = \frac{1}{2}\left(U_{n+m-1} + U_{n-m-1}\right) & \text{for all integers } 0 \leqslant m < n \end{cases}$$

The Chebyshev polynomials of the first and second kind are defined by $T_n(\cos\theta) = \cos(n\theta)$ and $U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$ for $\theta \notin \pi\mathbb{Z}$.

Show that
$$\begin{cases} T_m \cdot T_n = \frac{1}{2}\left(T_{n+m} + T_{n-m}\right) & \text{for all integers } 0 \leqslant m \leqslant n \\ T_m \cdot U_{n-1} = \frac{1}{2}\left(U_{n+m-1} + U_{n-m-1}\right) & \text{for all integers } 0 \leqslant m < n \end{cases}$$

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.C.1 QIII.C.2 QIV.A QIV.B QIV.C.1 QIV.C.2 QIV.C.3 QIV.C.4