Definition:Gaussian Binomial Coefficient

From ProofWiki
Jump to: navigation, search


Let $q \in \R_{\ne 1}$, $r \in \R$, $m \in \Z_{\ge 0}$.

The Gaussian binomial coefficient is an extension of the more conventional binomial coefficient as follows:

\(\displaystyle \binom r m_q\) \(:=\) \(\displaystyle \prod_{k \mathop = 0}^{m - 1} \dfrac {1 - q^{r - k} } {1 - q^{k + 1} }\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\left({1 - q^r}\right) \left({1 - q^{r - 1} }\right) \cdots \left({1 - q^{r - m + 1} }\right)} {\left({1 - q^m}\right) \left({1 - q^{m - 1} }\right) \cdots \left({1 - q^1}\right)}\) $\quad$ $\quad$

Also known as

Some sources refer to this concept as a $q$-nomial coefficient.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$'s view is that by referring to a construct by the specific names of the variables in which it is stated limits its flexibility of expression.

Also see

  • Results about Gaussian binomial coefficients can be found here.

Source of Name

This entry was named for Carl Friedrich Gauss.