Resources for arithmetic in quantum integers.

Question

I'm using the definition $$[n]=\frac{A^{2n}-A^{-2n}}{A^2-A^{-2}},$$ although the usual definition is recovered by substituting $q=A^2$.

In trying to prove various identities (and also reading the literature) I've had a lot of problems passing between the LHS and RHS of the top equation. I know that there are a lot of identities out there (I'm not sure where either) but I'm moreso looking for a program/ paper that collects this information.

For example, one can use a computer algebra system (I used sage) that can "prove" an identity, but what is returned is some mess in the RHS notation, rather than something meaningful with the quantum integers. I'm unfamiliar with manipulating them.

A successful program in my mind, would be able to translate $$\frac{A^{-1/4}}{A^{1/4}+A^{-1/4}}$$ into a reasonable closed form in quantum integer notation, since I struggle even to get these types of expressions going.

Answer 1

The product/quotient of quantum integers can be factored into homogeneous cyclotomic polynomials.

For example: $$\Phi_3(q)=[3],\; \Phi_4(q)=\frac{[4]}{[2]},\; \Phi_5(q)=[5],\; \Phi_6(q)=\frac{[6][1]}{[2][3]},\; \Phi_7(q)=[7],\;\Phi_8(q)= \frac{[8]}{[4]}.$$ A computer algebra system should be able to factor a product/quotient of quantum integers into cyclotomic polynomial factors.

As for list of identities, there is one source I know of. My list of Simple Algebraic Identities has many identities that are satisfied by trigonometric sine (indicated by $\texttt{[TS]}$ tag). For example: $$ \texttt{id2_3_1_2a = +a*a -b*b -(a-b)*(a+b)}.$$ This corresponds to the trigonometric identity $$ 0 = \sin(a)\sin(a) - \sin(b)\sin(b) - \sin(a-b)\sin(a+b),$$ but also the equivalent quantum integer identity $$ 0 = [n][n] - [m][m] - [n-m][n+m].$$

Answer 2

With your CAS you obtain a rational function in $A^{2/m}$ $$f(A^{2/m})=f(A^{-2/m}),\qquad f(x) = \frac{g(x)}{h(x)}, \qquad g,h \in \mathbb{C}[x]$$

$$f(x) = \frac{g(x)h(x^{-1})+g(x^{-1})h(x)}{2h(x)h(x^{-1})} = \frac{\sum_{n=0}^N b_n (x^n+x^{-n})}{\sum_{n=0}^M c_n (x^n+x^{-n})}$$ $$f(A^{2/m}) =\frac{\sum_{n=0}^N b_n \frac{[2n/m]}{[n/m]}}{\sum_{n=0}^M c_n \frac{[2n/m]}{[n/m]}}, \qquad [n/m] = \frac{A^{2n/m}-A^{-2n/m}}{A^{2}-A^{-2}}$$

That you can try to simplify with some other rules.