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 $x \in \mathbb{R}$.
Show that, for all $n \in \mathbb{N}^*$ and all real $x$, the following relation holds:
$$\left(1 - x^2\right) T_n''(x) - x T_n'(x) + n^2 T_n(x) = 0.$$