adding the ring of quantum-valued polynomials

[fchapoton]

This is a q-analogue of the ring of integer-valued polynomials. The elements are polynomials in x such that their evaluation at every q-integer is a polynomial in q.

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[tscrim]

I think it would be good to have q be included as an argument, subsequently it would be a part of the ground ring. Now you can have an option like if the input is :

1. Only R, then append q as currently implemented.
2. Both (R, q), then do q = R(q).
3. Just q and get R = q.parent().

[fchapoton]

Hmm, I would rather not allow a custom q. The ring R is not there to provide q, in my mind. I have added a sentence about that in the doc.

[tscrim]

You don’t expect anyone to want to specialize q? It also feels slightly off mathematically to not have q in the base ring, but perhaps that is just from my experience with Hecke algebras and quantum groups...

[fchapoton]

I am not sure. For my own usage, I have a base ring of coefficients, almost always QQ, and the variable q on top of that. For this algebra of quantum-valued polynomials, I can see no interest to have q a complex number. Of course, one can evaluate to polynomials and then evaluate q if one will.

[tscrim]

I believe certain Askey scheme $q$-orthogonal polynomials (e.g., cont. q-Hermite) would be in this ring, and problems I've worked on, we want to consider $q \in \mathbb{R}_{>0}$. There might also be important considerations for doing things over finite fields (or $q$ satisfying some relation). I think it would be better to be more general as it is easy to add and keep the most common behavior ($q$ is a generic parameter).

[tscrim]

I believe certain Askey scheme $q$-orthogonal polynomials (e.g., cont. q-Hermite) would be in this ring, and problems I've worked on, we want to consider $q \in \mathbb{R}_{>0}$. There might also be important considerations for doing things over finite fields (or $q$ satisfying some relation). I think it would be better to be more general as it is easy to add and keep the most common behavior ($q$ is a generic parameter).

[tscrim]

Can you also add the definition to this documentation?