Negated Upper Index of Gaussian Binomial Coefficient

Theorem
Let $q \in \R_{\ne 1}, r \in \R, k \in \Z$.

Then:
 * $\dbinom r k_q = \paren {-1}^k \dbinom {k - r - 1} k_q q^{k r - k \paren {k - 1} / 2}$

where $\dbinom r k_q$ is a binomial coefficient.

Proof
First note that:

Then: