Exercise 53, § 1

Let $u = 1 + T + \sum_{n=2}^\infty a_n T^n$ be an element of $\Lambda(\mathbf{Z})$. There exists one and only one group homomorphism $\lambda^u : \mathbf{Z} \to \Lambda(\mathbf{Z})$ such that $\lambda^u(1) = u$, and $(\mathbf{Z}, \lambda^u)$ is a pre-$\lambda$-ring. One has $\lambda^u(n) = u^n$ whatever $n \in \mathbf{Z}$. In order that $(\mathbf{Z}, \lambda^u)$ be a $\lambda$-ring, it is necessary and sufficient that $u = 1 + T$, in which case one has

whatever $n \in \mathbf{Z}$ and $m \in \mathbf{N}$.