In the fields of representation theory and quantum algebra, we often start with some $\mathbf{C}$-algebra and study it's quantization as an algebra over $\mathbf{C}(q)$, using the algebra structure to twist the multiplication. But what is $q$? Taking the calculus-flavored approach like in Quantum Calculus by Kac and Cheung, $q$ is just some indeterminate complex number: we specialize $q \to 1$ to get our original un-quantized object back, or we need $|q|<1$ for certain series to converge, or we specialize $q$ to some root of unity to make $q$ have finite order.
But in other applications, like counting subspaces of a vector space over $F_q$, or defining the structure constants of the Hall algebra of a $F_q$-linear category, we use the same $q$-calculus framework, but we are now thinking of $q$ as a power of a prime.
How do we consolidate the calculus interpretation of $q$ as an element of our base field with the combinatorial interpretation of $q$ as a prime power?
