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 $F_{a,c}(x)=F_{a,b}(x)F_{b,c}(x)$.
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asked Dec 2, 2015 at 19:31
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1$\begingroup$ 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). $\endgroup$– Johann CiglerCommented May 19, 2017 at 12:41
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$\begingroup$ @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? $\endgroup$– Fedor PetrovCommented May 19, 2017 at 15:40
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1 Answer
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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. ).
