# Definition:Gaussian Binomial Coefficient

## Definition

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.

## Sources

- 1971: Donald E. Knuth:
*Subspaces, Subsets, and Partitions*(*J. Combin. Th.***Ser. A****Vol. 10**: 178 – 180)

- 1990: George Gasper and Mizan Rahman:
*Basic Hypergeometric Series*

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(40)$