Using the results of the previous questions, conclude that $$\forall n \in \mathbb{N}^*, \quad \ell_n(\alpha X) = \sum_{k=1}^n \binom{n-1}{k-1} \alpha^k (1-\alpha)^{n-k} \ell_k(X)$$
Using the results of the previous questions, conclude that
$$\forall n \in \mathbb{N}^*, \quad \ell_n(\alpha X) = \sum_{k=1}^n \binom{n-1}{k-1} \alpha^k (1-\alpha)^{n-k} \ell_k(X)$$