Let $Q$ be a delta endomorphism with $Q = D \circ U$ where $U$ is the unique shift-invariant and invertible endomorphism. We denote by $(q_n)_{n \in \mathbb{N}}$ the sequence of polynomials associated with $Q$.
Deduce that, for all $n \in \mathbb{N}^*$,
$$n! q_n(X) = X U^{-n}\left(X^{n-1}\right)$$
then that
$$n q_n(X) = X (Q')^{-1}\left(q_{n-1}\right)$$