Negated Upper Index of Gaussian Binomial Coefficient

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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:

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


Then:

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

$\blacksquare$


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