Science is the belief in the ignorance of experts. —R. Feynman

]]>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 and ), 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 .

To continue reading, follow this link (pdf).

Etingof, Frenkel, Kirillov