I wonder whether there is a notation for such thing, which I denote $[a;b]\_q^n$ for a moment:
$$
[a;b]\_q^n:=(a-b)(a-qb)\dots (a-q^{n-1}b)=a^n(b/a;q)\_n,
$$
this last equation uses [$q$-Pochhammer symbol](https://en.wikipedia.org/wiki/Q-Pochhammer_symbol) notation.
My motivation is that it is a natural analogue of $(a-b)^n$ and some identities look better with it, at least annoying (well, this is subjective) powers of $q$ disappear. Say, [$q$-Vandermonde identity](https://en.wikipedia.org/wiki/Q-Vandermonde_identity) becomes
$$
[a;c]\_q^n=\sum {\binom{n}{k}}\_q[a;b]\_q^k\,[b;c]\_q^{n-k},
$$
compare to
$$
(a-c)^n=\sum \binom{n}{k} (a-b)^k (b-c)^{n-k}.
$$
Equivalently, for $q$-exponential generating functions $$F\_{a,b}(x):=\sum \frac{[a;b]\_q^k}{(k!)\_q}$$we have
$F\_{a,c}(x)=F\_{a,b}(x)F\_{b,c}(x)$.



notation for $(a-b)(a-qb)\dots (a-q^{n-1}b)$
There are some different notations in the literature: 
[B.A. Kupershmidt](http://de.arxiv.org/pdf/math/0004187.pdf) used $ {(a\dot - b)^n} $, 


Victor Kac and Pokman Cheung in “Quantum Calculus” used $(a-b)\_q^n$. 


In my [lecture notes](https://homepage.univie.ac.at/johann.cigler/skripten.html) I used ${(a\underset{\raise 0.5 em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{ - } b)^n}$ (with a dot below the minus sign. ).


