grandes-ecoles 2018 Q15

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

We denote by $n$ the integer part of $\frac{N}{2}$. We have $R_N(X) = U_N(X)^2$. We now assume that $U_{N}$ is odd.
For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$, and $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
Express $a_{N}$ again in terms of the $g_{\ell}$.

We denote by $n$ the integer part of $\frac{N}{2}$. We have $R_N(X) = U_N(X)^2$. We now assume that $U_{N}$ is odd.

For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$, and $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.

Express $a_{N}$ again in terms of the $g_{\ell}$.

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