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 $k \in \mathbb { N } ^ { * }$, $H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$. Let $\Delta _ { n }$ be the endomorphism induced by $\Delta$ on the stable subspace $\mathbb { R } _ { n } [ X ]$. Determine the matrix $A$ of $\Delta _ { n }$ in the basis $( H _ { 0 } , \ldots , H _ { n } )$.
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 $k \in \mathbb { N } ^ { * }$, $H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$.
Let $\Delta _ { n }$ be the endomorphism induced by $\Delta$ on the stable subspace $\mathbb { R } _ { n } [ X ]$. Determine the matrix $A$ of $\Delta _ { n }$ in the basis $( H _ { 0 } , \ldots , H _ { n } )$.