q-analog

Remarks

Example: $[n]_q$ is a q-analog of $n$, since $1 + q + \cdots + q^{n-1} \to n$ as $q \to 1$. Likewise $[n]_q!$ is a q-analog of $n!$. It is the generating function of the number of inversions over the permutations of $\set{1, \ldots, n}$: $[n]_q! = \sum_{\pi} q^{\operatorname{inv}(\pi)}$. For $n = 3$, the inversion numbers are $0, 1, 1, 2, 2, 3$, and $[3]_q! = (1)(1 + q)(1 + q + q^2) = 1 + 2q + 2q^2 + q^3$.

Non-example: $q^n$ is not a q-analog of $n$, since $q^n \to 1$ as $q \to 1$, and $1 \ne n$ for $n \ne 1$.

The definition by a limit alone does not fix a unique q-analog. In practice a q-analog is chosen because it keeps the structure of the original, such as a recurrence or a combinatorial meaning. Main examples are the Gaussian binomial coefficients, the q-Pochhammer symbol and the q-binomial theorem.