For $n \in \mathbb{Z}_{\geq 0}$, let $[n]_q := (1-q^n)/(1-q) = (1+q+...+q^{n-1})$ as is customary, with $[0]_q=0$.

Let $R$ be the subring of $\mathbb{Q}[q,q^{-1}][x]$ consisting of all $f$ such that $f([n]_q) \in \mathbb{Z}[q]$ for all $n \in \mathbb{Z}_{\geq 0}$.

Define $f_0(x) = 1$ and $f_k(x) = f_{k-1}(x)\cdot \frac{x-[k-1]_q}{q^{k-1}[k]_q}$ for $k \geq 1$. Note $f_k([n]_q) = \frac{[n]_q}{[n-k]_q [k]_q}=\binom{n}{k}_q \in \mathbb{Z}[q]$, so indeed $f_k \in R$ for all $k \geq 0$.

Is it the case that $R$ is spanned as a $\mathbb{Z}[q]$-module by the $f_k$? If so, what is known about the structure constants $c_{ij}^{k}(q)$?