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:

\(\ds \binom r m_q\)

\(:=\)

\(\ds \prod_{k \mathop = 0}^{m - 1} \dfrac {1 - q^{r - k} } {1 - q^{k + 1} }\)

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {1 - q^r} \paren {1 - q^{r - 1} } \cdots \paren {1 - q^{r - m + 1} } } {\paren {1 - q^m} \paren {1 - q^{m - 1} } \cdots \paren {1 - q^1} }\)

Also known as

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

Some use $q$-binomial 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: pp. 178 – 180)

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): binomial coefficient: 2.
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): $q$-binomial
• 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: $(40)$