grandes-ecoles 2016 QIII.C.2

grandes-ecoles · France · centrale-maths2__pc Proof Deduction or Consequence from Prior Results

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Deduce that $H_n(\mathbb{Z}) \subset \mathbb{Z}$, that is, $H_n$ is integer-valued on the integers.

We consider the family of polynomials
$$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$

Deduce that $H_n(\mathbb{Z}) \subset \mathbb{Z}$, that is, $H_n$ is integer-valued on the integers.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.A.6 QI.A.7 QI.A.8 QI.A.9 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QI.B.5 QI.B.6 QI.B.7 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.C QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.A.5 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.C.5 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C.1 QIV.C.2 QIV.C.3