Theorem 1

The map which to every formal series $u(S) = \sum_{k=0}^{\infty} \alpha_k S^k$ in one indeterminate over K associates the composition operator $u(D) = \sum_{k=0}^{\infty} \alpha_k D^k$ on $K[X]$ is an isomorphism of the algebra $K[[S]]$ of formal series onto the algebra $\Gamma$ of composition operators.

One verifies immediately that this map is an homomorphism. It remains to see that it is injective, in other words, that the relation $\sum_{k=0}^{\infty} \alpha_k D^k = 0$ implies $\alpha_k = 0$ for every $k$; now $h! \alpha_h$ is the constant term in the polynomial obtained by applying $\sum_{k=0}^{\infty} \alpha_k D^k$ to $X^h$, whence the theorem.