q-analog - "Classical" q-theory

"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 q-number of 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 Sn 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: $binom{n}{k}_q = frac{_q!}{_q! _q!}.$

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