For $p \in \mathbb{N}^*$, let $P$ be a non-zero polynomial solution of $(E_p) : x(y'' - y') + py = 0$. It has been shown that $U(f)$ satisfies $y'' - y' = -f(x)/x$ and that $U$ is self-adjoint ($\langle f | U(g) \rangle = \langle U(f) | g \rangle$). Prove that the function $pU(P) - P$ satisfies on $\mathbb { R } _ { + } ^ { * }$ the differential equation $y ^ { \prime \prime } - y ^ { \prime } = 0$.