13. Let $\mathscr { B } = \left( a _ { 0 } , \ldots , a _ { n } \right)$ be the unique orthogonal basis of $\left( \mathbb { R } _ { n } [ X ] , \langle \cdot , \cdot \rangle \right)$ such that $a _ { i }$ is a monic polynomial of degree $i$ for all $0 \leqslant i \leqslant n$. Show that, for all $0 \leqslant j \leqslant n - 1$, the coefficients of the polynomial $\prod _ { \ell = j + 1 } ^ { n } \left( X - r _ { \ell } \right)$ in the basis $\mathscr { B }$ are strictly positive real numbers.\\
Hint: one may denote $\left( q _ { j , 0 } , \ldots , q _ { j , n - j } \right)$ the basis of $\left( \mathbb { R } _ { n - j } [ X ] , \langle \cdot , \cdot \rangle _ { j } \right)$ obtained in questions 8a and 8b and reason by descending induction on $j$.

\section*{Third Part}
Let $\lambda$ be a strictly positive real number. For all real $x$ and $r$ such that $| x | < 1$ and $| r | < 1$, we set

$$F _ { \lambda } ( x , r ) = \left( 1 - 2 r x + r ^ { 2 } \right) ^ { - \lambda }$$

Show that the function $F _ { \lambda }$ is of class $\mathscr { C } ^ { \infty }$ on $] - 1,1 \left[ ^ { 2 } \right.$.\\