We apply the results of question 40 to the endomorphism $L$ defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$. We denote by $(\ell_n)_{n \in \mathbb{N}}$ its associated sequence of polynomials.
Verify that, for $n \in \mathbb{N}^*$,
$$\ell_n' = \ell_{n-1}' - \ell_{n-1}$$
and
$$X\ell_n'' - X\ell_n' + n\ell_n = 0$$
and
$$\ell_n(X) = \sum_{k=1}^n (-1)^k \binom{n-1}{k-1} \frac{X^k}{k!}$$