We denote by $n$ the integer part of $\frac{N}{2}$. We have $R_N(X) = U_N(X)^2$. We assume in this question that $U_{N}$ is even; we thus have $U_{N} \in \Pi_{n}$. In $\Pi_{n}$, the equation $P(1) = 1$ defines an affine subspace denoted $H_{n}$.
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 $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
(a) Show that $$\left\|U_{N}\right\|_{2} = \min\left\{\|P\|_{2} \mid P \in H_{n}\right\}$$
(b) Deduce that there exists a real number $\mu$ such that for all integers $0 \leqslant j \leqslant \frac{n}{2}$, we have $\left\langle U_{N}, P_{2j} \right\rangle = \mu$. (One may consider polynomials $P \in H_{n}$ of the form $U_{N} + t\left(P_{2j} - P_{2k}\right)$ with $t \in \mathbb{R}$.)
(c) Express $U_{N}$ in the basis of $P_{2j}$. Deduce that $$\frac{1}{\mu} = \sum_{0 \leqslant j \leqslant \frac{n}{2}} \frac{1}{g_{2j}}$$
(d) Establish in this case the formula $$a_{N} = \left(\sum_{0 \leqslant j \leqslant \frac{n}{2}} \frac{1}{g_{2j}}\right)^{-1}.$$