We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$. We denote $$g_{j} = \int_{-1}^{1} P_{j}(x)^{2}\,dx, \quad I_{j} = \int_{-1}^{1} \left(1 - x^{2}\right)^{j}\,dx$$ (a) Establish a relation between $g_{j}$ and $I_{j}$. (b) Find a relation between $I_{j}$ and $I_{j-1} - I_{j}$, and deduce a recurrence relation for the sequence $\left(I_{j}\right)_{j \in \mathbb{N}}$. (c) Deduce the value of $I_{j}$, then that of $g_{j}$.
We equip $\mathbb{R}_{n}[X]$ with the inner product defined by
$$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$
For $j \in \mathbb{N}$, we define the polynomial
$$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$
By convention, $P_{0} = 1$.
We denote
$$g_{j} = \int_{-1}^{1} P_{j}(x)^{2}\,dx, \quad I_{j} = \int_{-1}^{1} \left(1 - x^{2}\right)^{j}\,dx$$
(a) Establish a relation between $g_{j}$ and $I_{j}$.
(b) Find a relation between $I_{j}$ and $I_{j-1} - I_{j}$, and deduce a recurrence relation for the sequence $\left(I_{j}\right)_{j \in \mathbb{N}}$.
(c) Deduce the value of $I_{j}$, then that of $g_{j}$.