Reflection Rule for Gaussian Binomial Coefficients

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