Quantum (q-analog) numbers

Stream: Is there code for X?

[Robert Spencer]

Do we have statements and any results for q-numbers? These are (in some normalisations) $[n] = \frac{q^{n}-q^{-n}}{q-q^{-1}}.$ in the ring $\mathbb{Z}[q + q^{-1}]$.

[Kim Morrison]

I would love to see some of this done! You should look around for existing material on Pochhammer, but I think there is no quantum stuff yet.

[Kim Morrison]

Unfortunately there are many conventions in this world.

[Kim Morrison]

My bias is to use the representations theorists conventions (mostly as you've written above, balanced Laurent polynomials), rather than the combinatorialist conventions (which tend to be honest polynomials).

[Robert Spencer]

Sounds like a good place to start then. I agree regarding the convention of balanced Laurent polynomials, but might actually propose to work as much with the recursive definition of $[n] = [2][n-1] - [n-2]$ in $\mathbb{Z}[X]$ under $[2] = X$ as possible.

[Kim Morrison]

Chebyshev polynomials might already be out there, actually!