The q-factorial is the q-analog of the factorial function. It is written \([n]_q!\) or \(\mathrm{faq}(n,q)\) and is defined as \[[n]_q! = \prod^{n - 1}_{i = 0} \left( extstyle\sum^{i}_{j = 0} q^jight) = q^0 \cdot \left(q^0 + q^1ight) \cdot \left(q^0 + q^1 + q^2ight) \cdot \ldots \cdot \left(q^0 + q^1 + \ldots + q^{n - 1}ight)\] As with all q-analogs, letting \(q = 1\) produces the ordinary factorial. Based on the q-factorial, we can define the q-exponential function: \[e^x_q = \sum_{i = 0}^{\infty} \frac{x^i}{[i]_q!} = \frac{1}{[0]_q!} + \frac{x}{[1]_q!} + \frac{x^2}{[2]_q!} + \frac{x^3}{[3]_q!} + \cdots\]