# notation for $(a-b)(a-qb)\dots (a-q^{n-1}b)$

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 $F_{a,c}(x)=F_{a,b}(x)F_{b,c}(x)$.

• There are some notations in the literature. B. A. Kupershmidt (arXiv: math0004187) has used $(a-b)^n$ with a point above the minus sign (a don’t know how to write this in LaTex), I think also Kac and Cheung in Quantum Calculus use a similar notation (I do not have a copy to verify) and I have done the same with a point below the minus sign in my lecture notes "Elementare q-Identitaeten" (homepage.univie.ac.at/johann.cigler/skripten.html). – Johann Cigler May 19 '17 at 12:41
• @JohannCigler great, exactly what I was asking about. Would you please make it an answer so that I may accept it and question becomes answered? – Fedor Petrov May 19 '17 at 15:40

## 1 Answer

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. ).