TITLE: notation for $(a-b)(a-qb)\dots (a-q^{n-1}b)$
QUESTION [8 upvotes]: 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 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 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
$F_{a,c}(x)=F_{a,b}(x)F_{b,c}(x)$.

REPLY [3 votes]: There are some different notations in the literature: 
B.A. Kupershmidt  used $ {(a\dot  - b)^n} $, 
Victor Kac and Pokman Cheung in “Quantum Calculus” used $(a-b)_q^n$. 
In my lecture notes I used ${(a\underset{\raise 0.5 em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{ - } b)^n}$ (with a dot below the minus sign. ).