I'm looking at quantum calculus and just trying to understand what is going with this subject. Looking at the q-factorial made me wonder if this function could take all real or even complex numbers in the same way that $\Gamma (z)$ works as an extension of $f(n) =n!$. Since, I need practice with both $\Gamma $ and q-analogs, would it be a good project to try to recreate $\Gamma (z)$ in this new setting or is the whole project lacking in sanity, mathematical soundness?
Also, a minor question, why is there sometimes a coefficient in q-analog expansions as in this expression:
$(a;q)_n = \prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}).$
I'm a bit embarrassed not to know, but non of the lit. I have explains it, I'll just randomly see it tossed in there from time to time... and it really throws me off.
Thank you.