I am curious for what values of $\lambda \in \mathbb{R}^+$, the function $f(x)=\frac{e^{-\lambda } \lambda ^x}{x!}$ defined only on positive integers i-e $x \in \mathbb{Z}^+$ is invertible?
When $\lambda$ is any positive integer then there exists exactly one point in range of $f$ that is not invertible otherwise $f$ is one-to-one mapping ? But what can be said about rational and irrational values of $\lambda$ ?