I have been recently introduced to $q$-analogs, and find them quite interesting, specially for their connections going from Ramanujan expressions to very broad aplicable subjects.
As I humbly understand them they are parametrized families that give us regular combinatorial expressions when $q \rightarrow 1$, but that evaluated at other $q$ ( I think only integers) gave us counting formulas too, They also happen to interact very nicely between them. (factorials, binommials properties etc)
Then has anyone studied parametrized families that approach the regular expression when $ q \rightarrow 2$? Or when $q \rightarrow k$ for $k\in \mathbf{Z}$? Is $k=1$ the only interesting case?
Also Trying to not only play with integers, what if we modify exponential generating functions replacing $e$ with $e^q$, this would give us the regular result as $q \rightarrow 1$ again, but maybe give us some generalizations where $q$ is a parameter. Have this being studied? Exponential Generating functions for $q$ analogs too?
I know this is very open question but I just would like to speak a little about this with someone in this pandemic contrived academia, some expressions or just references could do.