We wish to show that, for every delta endomorphism $Q$, there exists a unique sequence of polynomials $(q_n)_{n \in \mathbb{N}}$ of $\mathbb{K}[X]$ such that:
$q_0 = 1$;
$\forall n \in \mathbb{N}, \deg(q_n) = n$;
$\forall n \in \mathbb{N}^*, q_n(0) = 0$;
$\forall n \in \mathbb{N}^*, Q q_n = q_{n-1}$.
Let $Q$ be a delta endomorphism. Show the existence and uniqueness of the sequence $(q_n)_{n \in \mathbb{N}}$ of polynomials associated with $Q$.
We wish to show that, for every delta endomorphism $Q$, there exists a unique sequence of polynomials $(q_n)_{n \in \mathbb{N}}$ of $\mathbb{K}[X]$ such that:
\begin{itemize}
\item $q_0 = 1$;
\item $\forall n \in \mathbb{N}, \deg(q_n) = n$;
\item $\forall n \in \mathbb{N}^*, q_n(0) = 0$;
\item $\forall n \in \mathbb{N}^*, Q q_n = q_{n-1}$.
\end{itemize}
Let $Q$ be a delta endomorphism. Show the existence and uniqueness of the sequence $(q_n)_{n \in \mathbb{N}}$ of polynomials associated with $Q$.