# The q-binomial coefficients and their interpretations

A q-derivative is a discrete version of the usual derivative, defined by the formula
$D_q f(x) = \frac{f(qx) - f(x)} {qx - x}.$

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 ${n \brack k }$ 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 $x=k$ and $y=0$), for every A (like a generating function);

(2) the number of subspaces of dimension $k$ in certain finite vector spaces: $\mathbb{F}_q^n$.

Finally, we point to some open questions concerning the construction of a $\mathbb F_1$.

“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