There seems to be a result for formal series (I hope this is right) for all integer $r\ge 0$ $$ \sum_{n\ge 0} (-x)^n\ {{n+r}\choose{r}}_{q} = (1+x)^{-1}(1+qx)^{-1}\dots (1+q^{r}x)^{-1} $$ where the $q$-binomial coefficients are written in terms of the $q$ integers $[t]=1+q+\dots+ q^{t-1}$. This is a fairly direct analogue of the classical negative binomial theorem (careful choice of $x$ will make it convergent).

I was hoping that someone could give a reference for where to find this - it does not seem to be trivial to prove. Apologies for not knowing what is likely a very standard reference!