If $T$ is an endomorphism of $\mathbb{K}[X]$, its Pincherle derivative $T'$ is defined by
$$\forall p \in \mathbb{K}[X], \quad T'(p) = T(Xp) - XT(p)$$
Show that, if there exists $(a_n)_{n \in \mathbb{N}}$ a sequence of scalars such that $T = \sum_{k=0}^{+\infty} a_k D^k$, then $T' = \sum_{k=1}^{+\infty} k a_k D^{k-1}$.