Let $\varphi$ be the function defined by $$\forall t \in ] - 1,1 \left[ \backslash \{ 0 \} , \quad \varphi ( t ) = ( 1 - t ) ^ { 1 - 1 / t } \right.$$ We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by $$\left\{ \begin{array} { l }
b _ { 0 } = - 1 \\
\forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k }
\end{array} \right.$$ Prove that, for all $t$ in $] - 1,1 \left[ , \varphi ^ { \prime } ( t ) = \varphi ( t ) \psi ( t ) \right.$, where $$\forall t \in ] - 1,1 \left[ , \quad \psi ( t ) = - \sum _ { n = 0 } ^ { + \infty } \frac { 1 } { n + 2 } t ^ { n } \right.$$ then that, for all $n$ in $\mathbb { N } ^ { * }$, $$\varphi ^ { ( n ) } ( 0 ) = - \sum _ { k = 0 } ^ { n - 1 } \frac { k ! } { k + 2 } \binom { n - 1 } { k } \varphi ^ { ( n - k - 1 ) } ( 0 )$$
Let $\varphi$ be the function defined by
$$\forall t \in ] - 1,1 \left[ \backslash \{ 0 \} , \quad \varphi ( t ) = ( 1 - t ) ^ { 1 - 1 / t } \right.$$
We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l }
b _ { 0 } = - 1 \\
\forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k }
\end{array} \right.$$
Prove that, for all $t$ in $] - 1,1 \left[ , \varphi ^ { \prime } ( t ) = \varphi ( t ) \psi ( t ) \right.$, where
$$\forall t \in ] - 1,1 \left[ , \quad \psi ( t ) = - \sum _ { n = 0 } ^ { + \infty } \frac { 1 } { n + 2 } t ^ { n } \right.$$
then that, for all $n$ in $\mathbb { N } ^ { * }$,
$$\varphi ^ { ( n ) } ( 0 ) = - \sum _ { k = 0 } ^ { n - 1 } \frac { k ! } { k + 2 } \binom { n - 1 } { k } \varphi ^ { ( n - k - 1 ) } ( 0 )$$