We denote $\alpha > -1/2$, $F_n$ the vector subspace of $E$ of polynomial functions of degree less than or equal to $n$ (where $n \in \mathbb{N}$),
$$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$
and
$$S_\alpha(f,g) = \int_{-1}^{1} f(t)g(t)\left(1-t^2\right)^{\alpha - \frac{1}{2}} \mathrm{~d}t$$
Justify that two eigenvectors of $\varphi_\alpha$ of distinct degrees are orthogonal (with respect to $S_\alpha$).