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 $8 a$ and $8 b$ 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

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

\begin{enumerate}
  \setcounter{enumi}{13}
  \item Show that the function $F _ { \lambda }$ is of class $\mathscr { C } ^ { \infty }$ on $] - 1,1 \left[ ^ { 2 } \right.$.
  \item Show that for $x \in ] - 1,1 \left[ \right.$, the function $r \mapsto F _ { \lambda } ( x , r )$ is expandable as a power series in a neighborhood of 0 .\\
For $x \in ] - 1,1 \left[ \right.$, we denote $a _ { n } ^ { ( \lambda ) } ( x )$ the $n$-th coefficient of the expansion of the function $r \mapsto F _ { \lambda } ( x , r )$ so that, for $r$ in a neighborhood of 0 ,
\end{enumerate}

$$F _ { \lambda } ( x , r ) = \sum _ { n \geqslant 0 } a _ { n } ^ { ( \lambda ) } ( x ) r ^ { n }$$

16a. For $x \in ] - 1,1 \left[ \right.$, show that $a _ { 1 } ^ { ( \lambda ) } ( x ) = 2 \lambda x a _ { 0 } ^ { ( \lambda ) } ( x )$ and that, for every integer $n \geqslant 1$,

$$( n + 1 ) a _ { n + 1 } ^ { ( \lambda ) } ( x ) = 2 ( n + \lambda ) x a _ { n } ^ { ( \lambda ) } ( x ) - ( n + 2 \lambda - 1 ) a _ { n - 1 } ^ { ( \lambda ) } ( x )$$

Hint: one may begin by computing $\left( 1 - 2 x r + r ^ { 2 } \right) \frac { \partial F _ { \lambda } } { \partial r } ( x , r )$.\\
16b. Deduce that, for all $n \geqslant 0$, the function $a _ { n } ^ { ( \lambda ) }$ is a polynomial of degree $n$ whose leading coefficient and parity will be determined.\\
We now assume that $\lambda > \frac { 1 } { 2 }$. For $P , Q \in \mathbb { R } [ X ]$, we set

$$\langle P , Q \rangle = \int _ { - 1 } ^ { 1 } P ( x ) Q ( x ) \left( 1 - x ^ { 2 } \right) ^ { \lambda - \frac { 1 } { 2 } } d x$$