Reflection Rule for Gaussian Binomial Coefficients

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Theorem

Let $q \in \R_{\ne 1}, n \in \Z_{>0}, k \in \Z$.

Then:

$\dbinom n k_q = q^{k \left({n - k}\right)} \dbinom n k_{q^{-1} }$

where $\dbinom n k_q$ is a Gaussian binomial coefficient.


Proof

\(\displaystyle \dbinom n k_{q^{-1} }\) \(=\) \(\displaystyle \prod_{j \mathop = 0}^{k - 1} \dfrac {1 - \left({q^{-1} }\right)^{n - j} } {1 - \left({q^{-1} }\right)^{j + 1} }\) Definition of Gaussian Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle \prod_{j \mathop = 0}^{k - 1} \dfrac {\frac {q^{n - j} - 1} {q^{n - j} } } {\frac {q^{j + 1} - 1} {q^{j + 1} } }\)
\(\displaystyle \) \(=\) \(\displaystyle \prod_{j \mathop = 0}^{k - 1} \dfrac {q^{j + 1} } {q^{n - j} } \dfrac {q^{n - j} - 1} {q^{j + 1} - 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \prod_{j \mathop = 0}^{k - 1} q^{1 - n} \dfrac {1 - q^{n - j} } {1 - q^{j + 1} }\)
\(\displaystyle \) \(=\) \(\displaystyle \left({q^{1 - n} }\right)^k \prod_{j \mathop = 0}^{k - 1} \dfrac {1 - q^{n - j} } {1 - q^{j + 1} }\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {q^{k \left({n - 1}\right)} } \dbinom n k_q\) Definition of Gaussian Binomial Coefficient



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