We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$ Show that, for all non-zero natural integer $p$, there exist two polynomials $P _ { p }$ and $Q _ { p }$ with real coefficients such that, for all $x \in ] - \infty , 1 [$, $$\varphi ^ { ( p ) } ( x ) = \frac { P _ { p } ( \sqrt { 1 - x } ) } { Q _ { p } ( \sqrt { 1 - x } ) } \exp \left( \frac { - x } { \sqrt { 1 - x } } \right)$$
We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by
$$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Show that, for all non-zero natural integer $p$, there exist two polynomials $P _ { p }$ and $Q _ { p }$ with real coefficients such that, for all $x \in ] - \infty , 1 [$,
$$\varphi ^ { ( p ) } ( x ) = \frac { P _ { p } ( \sqrt { 1 - x } ) } { Q _ { p } ( \sqrt { 1 - x } ) } \exp \left( \frac { - x } { \sqrt { 1 - x } } \right)$$