I'm considering a series of polynomial functions defined as follows: let $q\in \mathbb{R}$, $q\neq 0$. For each $n\geq 1$ let $$ f_n(x):=\prod_{i=1}^nq^ix(1-q^ix). $$ This series of functions looks like Hermite functions but not the same.
My question is: is the above function a well-known special function? If yes, where can I find its properties?
This is related to the $q$-Pochhammer Symbol, $$f_n(x):=\prod_{i=1}^nq^ix(1-q^ix)=q^{n(n+1)/2}x^n(qx;q)_{n}.$$