We fix $n \in \mathbb { N }$. We define the linear map: $$\begin{aligned}
\Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\
P ( X ) & \mapsto P ( X + 1 ) - P ( X )
\end{aligned}$$ We define $H _ { 0 } ( X ) = 1$ 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. Deduce that $U _ { n } ( p ) = \sum _ { k = 0 } ^ { n } \frac { S ( n , k ) } { k + 1 } H _ { k + 1 } ( p + 1 )$.
We fix $n \in \mathbb { N }$. We define the linear map:
$$\begin{aligned}
\Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\
P ( X ) & \mapsto P ( X + 1 ) - P ( X )
\end{aligned}$$
We define $H _ { 0 } ( X ) = 1$ 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.
Deduce that $U _ { n } ( p ) = \sum _ { k = 0 } ^ { n } \frac { S ( n , k ) } { k + 1 } H _ { k + 1 } ( p + 1 )$.