arithmetic with quantum integers

Question

Consider the ring $\mathbb{Z}[q^{\pm 1}]$. For $n \in \mathbb{N}$, define the quantum integers:

$$[n]_q := \frac{q^n-q^{-n}}{q-q^{-1}} = q^{n-1} + q^{n-3} + \cdots + q^{-(n-3)} + q^{-(n-1)}$$

What is the general formula for multiplying and dividing quantum integers? This is probably well-known but I don't have a reference. For example, we have

$$ [2]_q[2]_q = [3]_q+1, \ \ \ \ [4]_q[3]_q = [6]_q+[4]_q+[2]_q, \ \ \ \ \frac{[6]_q}{[2]_q} = [5]_q-[3]_q+1 $$

Also, what is the relationship between these quantum integers and the ones defined as

$$ [n]_q := \frac{1-q^n}{1-q} = 1 + q + \dots + q^{n-1}? $$

Thanks. Edit: fixed first formula

Answer 1

A bit long for a comment:

There's a "more standard" way of making the quantum integers into a ring via so-called quantum addition: $[x]_q\oplus_q[y]_q=[x]_q+q^x[y]_q$. If you work it out, this will give you $[x]_q\oplus_q [y]_q=[x+y]_q$.

There's a similar definition for multiplication: $[x]_q\otimes_q [y]_q=[x]_q[y]_{q^x}$, which if you work out gives you $[x]_q\otimes_q [y]_q=[xy]_q$. For more details, see the first few pages of this paper. Most importantly, they make the set of $[n]_q$ into a ring. In section 3 of the paper, they work out more classical results of $[x]_q[y]_q$ and $[x+y]_q$.

Answer 2

I worked out a formula for multiplication since asking this question:

$$ [n]_q[n+k]_q = \sum_{j=0}^{n-1} [2n-1+k-2j]_q $$

I haven't found a big need for a formula for division yet so haven't tried to work it out, but I'll update my answer if I find one. Though I do note that you don't generally end up with a linear combination of quantum integers, for example

$$ \frac{[5]_q}{[2]_q} = \frac{1+q^2+q^4+q^6+q^8}{q^3+q^5} = \frac{1}{q+q^{-1}}[5]_q $$

but it happens that in my case I am interested in precisely what divisions result in a linear combination of quantum integers. I'm still interested in any other references for quantum integers.