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 $k \in \mathbb { N } ^ { * }$, $$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$ and $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts. III.B.1) For all $k \in \mathbb { N }$, establish a simplified expression for $H _ { k + 1 } ( X ) + k H _ { k } ( X )$. III.B.2) Deduce that, for every natural integer $n$ $$X ^ { n } = \sum _ { k = 0 } ^ { n } S ( n , k ) H _ { k } ( X )$$
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 $k \in \mathbb { N } ^ { * }$,
$$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$
and $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
III.B.1) For all $k \in \mathbb { N }$, establish a simplified expression for $H _ { k + 1 } ( X ) + k H _ { k } ( X )$.
III.B.2) Deduce that, for every natural integer $n$
$$X ^ { n } = \sum _ { k = 0 } ^ { n } S ( n , k ) H _ { k } ( X )$$