15. 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 $\mathbf { 0 }$. For $x \in ] - 1,1$, 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 ,
$$F _ { \lambda } ( x , r ) = \sum _ { n \geqslant 0 } a _ { n } ^ { ( \lambda ) } ( x ) r ^ { n } .$$
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