We define the sequence of polynomials $\left( H _ { k } \right) _ { k \in \mathbb { N } }$ in $\mathbb { R } [ X ]$ by $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$ It has been established that for all $k \in \mathbb { N }$ and $x \in ] - 1,1 [$, $$\frac { \left( \mathrm { e } ^ { x } - 1 \right) ^ { k } } { k ! } = \sum _ { n = k } ^ { + \infty } S ( n , k ) \frac { x ^ { n } } { n ! }$$ III.D.1) For $x \in ] - 1,1 [$ and $\alpha \in \mathbb { R }$, simplify $\sum _ { k = 0 } ^ { + \infty } H _ { k } ( \alpha ) \frac { x ^ { k } } { k ! }$. III.D.2) Show that for $u < \ln 2$ $$\mathrm { e } ^ { u \alpha } = \sum _ { k = 0 } ^ { + \infty } H _ { k } ( \alpha ) \frac { \left( \mathrm { e } ^ { u } - 1 \right) ^ { k } } { k ! }$$
We define the sequence of polynomials $\left( H _ { k } \right) _ { k \in \mathbb { N } }$ in $\mathbb { R } [ X ]$ by $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$,
$$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$
It has been established that for all $k \in \mathbb { N }$ and $x \in ] - 1,1 [$,
$$\frac { \left( \mathrm { e } ^ { x } - 1 \right) ^ { k } } { k ! } = \sum _ { n = k } ^ { + \infty } S ( n , k ) \frac { x ^ { n } } { n ! }$$
III.D.1) For $x \in ] - 1,1 [$ and $\alpha \in \mathbb { R }$, simplify $\sum _ { k = 0 } ^ { + \infty } H _ { k } ( \alpha ) \frac { x ^ { k } } { k ! }$.
III.D.2) Show that for $u < \ln 2$
$$\mathrm { e } ^ { u \alpha } = \sum _ { k = 0 } ^ { + \infty } H _ { k } ( \alpha ) \frac { \left( \mathrm { e } ^ { u } - 1 \right) ^ { k } } { k ! }$$