# Negated Upper Index of Gaussian Binomial Coefficient

## Theorem

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

Then:

$\dbinom r k_q = \left({-1}\right)^k \dbinom {k - r - 1} k_q q^{k r - k \left({k - 1}\right) / 2}$

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

## Proof

First note that:

 $\ds 1 - q^t$ $=$ $\ds q^{-t} \dfrac {1 - q^t} {q^{-t} }$ $\ds$ $=$ $\ds \dfrac {q^{-t} - 1} {q^{-t} }$ $\ds$ $=$ $\ds q^t \left({q^{-t} - 1}\right)$ $\text {(1)}: \quad$ $\ds$ $=$ $\ds -q^t \left({1 - q^{-t} }\right)$

Then:

 $\ds \binom r k_q$ $=$ $\ds \prod_{j \mathop = 0}^{k - 1} \dfrac {1 - q^{r - j} } {1 - q^{j + 1} }$ Definition of Gaussian Binomial Coefficient $\ds$ $=$ $\ds \prod_{j \mathop = 0}^{k - 1} \dfrac {-q^{r - j} \left({1 - q^{-\left({r - j}\right)} }\right)} {1 - q^{j + 1} }$ from $(1)$ $\ds$ $=$ $\ds \left({-1}\right)^k \prod_{j \mathop = 0}^{k - 1} \dfrac {q^{r - j} \left({1 - q^{-\left({r - j}\right)} }\right)} {1 - q^{j + 1} }$ $\ds$ $=$ $\ds \left({-1}\right)^k \prod_{j \mathop = 0}^{k - 1} \dfrac {q^{-\left({j - r}\right)} \left({1 - q^{j - r} }\right)} {1 - q^{j + 1} }$ $\ds$ $=$ $\ds \left({-1}\right)^k \prod_{j \mathop = 0}^{k - 1} \dfrac {q^{-\left({\left({k - 1}\right) - j - r}\right)} \left({1 - q^{\left({k - 1}\right) - j - r} }\right)} {1 - q^{j + 1} }$ Permutation of Indices of Product $\ds$ $=$ $\ds \left({-1}\right)^k \prod_{j \mathop = 0}^{k - 1} \dfrac {1 - q^{\left({k - r - 1}\right) - j} } {1 - q^{j + 1} } \prod_{j \mathop = 0}^{k - 1} q^{-\left({\left({k - 1}\right) - j - r}\right)}$ $\ds$ $=$ $\ds \left({-1}\right)^k \prod_{j \mathop = 0}^{k - 1} \dfrac {1 - q^{\left({k - r - 1}\right) - j} } {1 - q^{j + 1} } \prod_{j \mathop = 1}^k q^j \prod_{j \mathop = 0}^{k - 1} q^{r - k}$ $\ds$ $=$ $\ds \left({\left({-1}\right)^k \prod_{j \mathop = 0}^{k - 1} \dfrac {1 - q^{k - r - 1 - j} } {1 - q^{j + 1} } }\right) q^{k r - k \left({k - 1}\right) / 2}$ Closed Form for Triangular Numbers and algebra $\ds$ $=$ $\ds \left({-1}\right)^k \dbinom {k - r - 1} k_q q^{k r - k \left({k - 1}\right) / 2}$ Definition of Gaussian Binomial Coefficient

$\blacksquare$