When dealing with an element $P \in \mathbb{Z}[q]$, a natural combinatorial question is to ask whether $P$ has non-negative coefficients. For instance, this is true if $P$ is a Kazhdan-Lusztig polynomial. I have recently been looking at certain elements of the field $\mathbb{Q}(q)$ that have a similar flavor to Kazhdan-Lusztig polynomials and have observed some positivity properties.

What is the correct definition of non-negativity in this ring?

At first I thought a natural candidate would be to say that $f \in \mathbb{Q}(q)$ is non-negative if $f = P/Q$, $P, Q \in \mathbb{Z}[q]$, $P$ and $Q$ have no common factors, and $P$ and $Q$ have non-negative coefficients. However, this is not a good definition because then the set of non-negative elements is not a semiring. For example, $[2],[3],$ and $\frac{1}{[6]}$ are all non-negative by this definition, but $\frac{[2][3]}{[6]} = \frac{q^2}{1-q^2+q^4}$ is not, where $[k] = \frac{q^k - q^{-k}}{q - q^{-1}}$.

Another possible definition of non-negativity would be $f \in \mathbb{Q}(q)$ is non-negative if $f(a)$ is non-negative for all non-negative real $a$. With this definition, the set of non-negative rational functions is a semiring. This is strictly weaker than the definition of non-negativity above.

Has anyone come across this definition of non-negativity before?