$[m]_q = \frac{q^m−q^{-m}}{q-q^{-1}}$ is the q- analogue of the number $m\in\mathbb{Z}_{\geq0}$.
$[0]_q=0$, $[1]_q=1$ and $[2]_q=q+q^{-1}$.
I don't know how to prove
- Every $[m]_q$ can be expressed as polynomial in $[2]_q$.
Indeed this is the case. $[3]_q=[2]^2_q-1$ and $[4]_q=[2]^3_q-2[2]_q$.
I don't know how to prove this in general.
Any reference to study the properties of q- analogue of the number will also be helpful.