For every $n \in \mathbb{N}$, define the polynomial $q_n = \frac{X^n}{n!}$. Let $T$ be an endomorphism of $\mathbb{K}[X]$. Show that $T$ is a shift-invariant endomorphism if, and only if, $$T = \sum_{k=0}^{+\infty} \left(T q_k\right)(0) D^k$$
For every $n \in \mathbb{N}$, define the polynomial $q_n = \frac{X^n}{n!}$. Let $T$ be an endomorphism of $\mathbb{K}[X]$.
Show that $T$ is a shift-invariant endomorphism if, and only if,
$$T = \sum_{k=0}^{+\infty} \left(T q_k\right)(0) D^k$$