# A $q$-analogue of a characterization of polynomials by binomial coefficients

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 $$[n]_q = 1+q+\cdots + q^{n-1}$$, $$[n]_q! = [n]_q[n-1]_q\ldots _q$$, 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.

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 $$\bigl( P(_q), P(_q), \ldots, P([N-1]_q) \bigr) \in \bigl\langle u^{(0)}_q, u^{(1)}_q, \ldots, u^{(d)}_q \bigr\rangle$$ if and only if $$\deg P\leqslant d$$.
• Thank you. I think the critical point is that the binomial coefficients $\binom{x}{d}$ are the minimal degree polynomials that interpolate the functions $0 \mapsto 0, 1 \mapsto 0, \ldots, d-1 \mapsto 0, d \mapsto 1$. Thus $(v_0, \ldots, v_{N-1})$ is in the span of $u_1^{(0)}, \ldots, u_1^{(d)}$ if and only if there is a polynomial $P$ of degree $\le d$ such that $P(j) = v_j$ for each $j$. This generalizes to the $q$-binomial coefficients replacing $j \mapsto 0$, $d \mapsto 1$ with $[j] \mapsto 0$ and $[d]_q \mapsto 1$ and $P(j) = v_j$ with $P([j]_q) = v_j$. Jul 16 '20 at 10:21
• Hence if $P(ax+b) = Q(x)$ then $(P(_q), \ldots, P([N-1]_q)) \in \langle u_q^{(0)}, \ldots, u_q^{(d)} \rangle$ if and only if $(Q(_q), \ldots, Q([N-1]_q)) \in \langle u_q^{(0}), \ldots, u_q^{(d)} \rangle$. For binomial coefficients this is useful, because such affine changes of variable move the evaluation points in a 'natural way'. (E.g. they can be reversed.) But for $q$-binomial coefficients, it's not so useful: e.g, there is no affine transformation sending $_1, _q, _q$ to $_q, _q, _q$ when $q\not= 1$. Jul 16 '20 at 10:26