$\frac{\binom{n}{j}_q\binom{n+1}{j}_q \cdots\binom{n+k-1}{j}_q}{\binom{j}{j}_q\binom{j+1}{j}_q\cdots\binom{j+k-1}{j}_q}$ where $n,j,k$ are non-negative integers.
Is the following product of $q$-binomial coefficients a polynomial in $q$?
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1It means it is$q$binomial coefficient and here is the link to its definition http://mathworld.wolfram.com/q-BinomialCoefficient.html – 2011-12-28
1 Answers
EDIT: I previously gave a flawed counterexample to the question, but I discovered a flaw in it and now believe the statement in the question to be true. The last two paragraphs are not a formal proof of this, but I believe could be turned into such a proof.
If $n
The factor $(q-\zeta_j)$ of $[j]_q$ cancels with the factor $(q-\zeta_{im}^i)$ of $[ij]_q$ chosen as the unique multiple of $j$ among $n,n-1,\ldots,n-j+1$. Similarly, the factor $(q-\zeta_j^2)$ cancels with $(q-\zeta_{im}^{2i})$, and repeating this procedure we cancel all the factors of $[j]_q$ with factors of $[ij]_q$. We can pick a multiple i'(j-1) of $j-1$ among $\{n,n-1,\ldots,n-j+1\}-\{ij\}$, and similarly a multiple of $j-2$ among \{n,n-1,\ldots,n-j+1\}-\{ij,i'(j-1)\}, and by repeating this procedure cancel all factors of $[j]_q[j-1]_q\cdots[1]_q$ with factors of $[n]_q[n-1]_q\cdots[n-j+1]_q$.
These are easy factors to cancel, because some multiple of $j$ is among $n+1,n,\ldots,n-j+1$. But what about the factors of $[j+1]_q$? Well, we have i'(j+1)=i'j+i' among $n,n-1,\ldots,n-j+2$ for some i', as the gap between i'(j+1) and (i'+1)(j+1) is $j+1$, and so we can cancel its factors with those of $[n-j+1]_q$ or of one of $[n+1]_q,\ldots,[n-j+2]_q$. But wait, we've already used $[n-j+1]_q$ to cancel factors in the denominator! True, but if $n-j+1$ is a multiple of $j+1$ then we must have used it to cancel factors of $[\frac{j+1}{2}]_q$ or smaller, so we can still cancel at least half the factors of $[j+1]_q$, and the other half will involve roots of unity of smaller order and so can be canceled with other leftovers among $[n]_q,\ldots,[n-j+1]_q$. The same should extend to $[j]_q,[j-1]_q,\ldots,[2]_q$. Iterating this argument should allow cancellation of all the factors in the denominator.