A q-derivative is a discrete version of the usual derivative, defined by the formula
This motivates a “quantum calculus” that appears in several branches of mathematics (mainly in number theory and in the theory of quantum groups).
Here, we present the basic q-deformed structures (q-derivative, q-integers, q-binomial coefficients), the basic theorem on Taylor series expansions and two “q-versions” of the binomial theorem.
The q-binomial coefficients
have two combinatorial interpretations:
(1) it counts the number of paths from (0,0) to (k,n-k) that encircle a prescribed area A (with respect to the axes
), for every A (like a generating function);
(2) the number of subspaces of dimension in certain finite vector spaces: .
Finally, we point to some open questions concerning the construction of a .
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“By that time, all three of us had already been severely afflicted with the “q-disease”, a dangerous mathematical illness whose earliest victim was Euler, but which was first diagnosed by Richard Askey. Mathematicians working in practically every field, be it algebra, geometry, analysis, differential equations -you name it- are vulnerable to its seductive charm. The first symptom of the q-disease is that one day you realize that most of the results obtained or acquired during your mathematical life admit a q-deformation. The second stage is indicated by the idea that the q-case is much more interesting.”
Etingof, Frenkel, Kirillov