Gaussian Binomial Theorem/Real Numbers

Theorem
Let $r \in \R$ be a real number.


 * $\displaystyle \sum_{k \mathop \in \Z} \dbinom r k_q q^{k \left({k - 1}\right) / 2} x^k = \prod_{k \mathop \ge 0} \dfrac {1 + q^k x} {1 + q^{r + k} x}$

where:
 * $\dbinom r k_q$ denotes a Gaussian binomial coefficient.
 * $x \in \R: \left\lvert{x}\right\rvert < 1$
 * $q \in \R: \left\lvert{q}\right\rvert < 1$.