We fix $\alpha > 0$, define $W : p \mapsto p(\alpha X)$, and set $P = W \circ L \circ W^{-1}$ where $L$ is the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$. We have $P = \frac{1}{\alpha} D \circ \left(\frac{1}{\alpha} D - I\right)^{-1}$.
Show that $P$ is a delta endomorphism whose associated polynomial sequence $(p_n)_{n \in \mathbb{N}}$ satisfies
$$\forall n \in \mathbb{N}, \quad p_n = \ell_n(\alpha X)$$