**"Classical" q-theory**

Classical *q*-theory begins with the *q*-analogs of the nonnegative integers. The equality

suggests that we define the *q*-analog of *n*, also known as the ** q-bracket** or

**of**

*q*-number*n*, to be

By itself, the choice of this particular *q*-analog among the many possible options is unmotivated. However, it appears naturally in several contexts. For example, having decided to use _{q} as the *q*-analog of *n*, one may define the *q*-analog of the factorial, known as the q-factorial, by

This *q*-analog appears naturally in several contexts. Notably, while *n*! counts the number of permutations of length *n*, we have that _{q}! counts permutations while keeping track of the number of inversions. That is, if inv(*w*) denotes the number of inversions of the permutation *w* and *S*_{n} denotes the set of permutations of length *n*, we have

In particular, one recovers the usual factorial by taking the limit as .

The *q*-factorial also has a concise definition in terms of the q-Pochammer symbol, a basic building-block of all *q*-theories:

- .

From the *q*-factorials, one can move on to define the ** q-binomial coefficients**, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients:

which allows q-addition to be defined:

and subtraction:

The q-exponential is defined as:

Q-trigonometric functions, along with a q-Fourier transform have been defined in this context

Read more about this topic: analog">*q*-analog

### Famous quotes containing the word classical:

“Several *classical* sayings that one likes to repeat had quite a different meaning from the ones later times attributed to them.”

—Johann Wolfgang Von Goethe (1749–1832)