# When $\frac{e^{-\lambda } \lambda ^x}{x!}$ over positive integers is invertible?

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 $\lambda$ ?

• What do you mean "invertible?" We rarely talk about "invertible" functions from $\mathbb N\to\mathbb R$. Do you mean one-to-one? – Thomas Andrews Nov 13 '15 at 11:44
• Yes one to one. – kaka Nov 13 '15 at 12:51
• Then the factor $e^{-\lambda}$ is a red herring. If $\lambda^n/n!$ is $1-1$ then so is $e^{-\lambda}\lambda^n/n!$, and visa versa. – Thomas Andrews Nov 13 '15 at 13:43
• But it does not answer my question. – kaka Nov 13 '15 at 13:57
• I'm sorry for trying to help you. I'll stop now. – Thomas Andrews Nov 13 '15 at 13:58