q-factorial

The q-factorial of $n$ is

[n]_{q}! = [1]_{q} \cdot [2]_{q} \cdot \dots \cdot [n]_{q} = \frac{(1 - q) (1 - q^{2}) \dots (1 - q^{n})}{(1 - q)^{n}}

where $[i]_{q}$ is a q-number.

Example:

Properties

Note that $(1 - q) ∣ 1 - q^{i}$ so this is always a polynomial
The coefficients are positive, symmetric, and unimodal
- the first two follows from this theorem