grandes-ecoles 2010 QI.A.5

grandes-ecoles · France · centrale-maths1__pc Taylor series Recursive or implicit derivative computation for series coefficients

Throughout the rest of this problem, we set $T_0(x) = 1$. For $n \in \mathbb{N}^*$, we denote by $T_n$ the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ for all $x \in \mathbb{R}$.
Determine two real numbers $a$ and $b$ such that $$\forall x \in \mathbb{R}, \forall n \in \mathbb{N}^*, T_{n+2}(x) = a x T_{n+1}(x) + b T_n(x)$$

Throughout the rest of this problem, we set $T_0(x) = 1$. For $n \in \mathbb{N}^*$, we denote by $T_n$ the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ for all $x \in \mathbb{R}$.

Determine two real numbers $a$ and $b$ such that
$$\forall x \in \mathbb{R}, \forall n \in \mathbb{N}^*, T_{n+2}(x) = a x T_{n+1}(x) + b T_n(x)$$

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.B.3 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B.1 QII.B.2 QII.B.3 QII.C QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2