Considering the binomial coefficient $\binom{x}{m}$ as a polynomial in $x$, the span of $\binom{x}{0}, \binom{x}{1}, \ldots, \binom{x}{d}$ is exactly the polynomials of degree $\le d$. A closely related characterization is that this subspace is the kernel of $\Delta^{d+1}$, where $\Delta : \mathbb{C}[x] \rightarrow \mathbb{C}[x]$ is the difference operator defined by $(\Delta P)(x) = P(x) - P(x-1)$. Roughly stated, my question asks for a $q$-analogue of either of these results.


To make this more concrete, use the standard notation, so $[n]\_q = 1+q+\cdots + q^{n-1}$, $[n]\_q! = [n]\_q[n-1]\_q\ldots [1]\_q$, and $\binom{n}{d}\_q = \frac{[n]\_q!}{[d]\_q![n-d]\_q!}$. Fix $N \in \mathbb{N}$ and for each $m \in \mathbb{N}\_0$ let


$$u^{(m)}\_q = \bigl( \binom{0}{m}\_q, \binom{1}{m}\_q, \ldots, \binom{N-1}{m}\_q \bigr) \in \mathbb{C}[q]^{N}$$


By the first paragraph, for $P \in \mathbb{C}[x]$, we have


$$\bigl( P(0), P(1), \ldots, P(N-1) \bigr) \in \bigl\langle u^{(0)}\_1, u^{(1)}\_1, \ldots, u^{(d)}\_1 \bigr\rangle$$


if and only if $\mathrm{deg} P \le d$. In a current research problem, it's useful that the same holds for $( P(N-1), \ldots, P(1), P(0))$; in fact the evaluation points $0$, $1, \ldots, N-1$ can be varied by an arbitrary affine transformation.



> 
> Is there an analogous characterization of the span $\langle u\_q^{(0)}, u\_q^{(1)}, \ldots, u\_q^{(d)} \rangle$?
> 
> 
> 


As a follow up, what transformations preserve this space? In particular, is it invariant under a $q$-analogue of the affine transformation just mentioned? Despite some thought I have not found any reasonable answer to these questions, but I find it hard to believe that there is nothing to be said.



A $q$-analogue of a characterization of polynomials by binomial coefficients
We have
$$\binom{n}d\_q=\frac{(q^n-1)\ldots(q^n-q^{d-1})}{(q^d-1)\ldots(q^d-q^{d-1})}=f\_d(q^n)=g\_d([n]\_q)
$$
where $f\_d$ and $g\_d$ are polynomials of degree $d$ (depending on $q$ of course). Therefore $$\bigl( P([0]\_q), P([1]\_q), \ldots, P([N-1]\_q) \bigr) \in \bigl\langle u^{(0)}\_q, u^{(1)}\_q, \ldots, u^{(d)}\_q \bigr\rangle$$
if and only if $\deg P\leqslant d$.


