高斯二项式系数

**高斯二項式係數** (也稱作 **高斯係數**, **高斯多項式**, 或 **_q_- 二項式係數**) 在數學里是指二項式係數的 q - 模擬。

## 定義

高斯二項式係數被定義為：

$$
{m \choose r}_{q}={\begin{cases}{\frac {(1-q^{m})(1-q^{m-1})\cdots (1-q^{m-r+1})}{(1-q)(1-q^{2})\cdots (1-q^{r})}}&r\leq m\\0&r>m\end{cases}}
$$

其中， _m_ 和 _r_ 是非負整數。 當 _r_ = 0 時值為 1。

高斯二項式係數計算一個有限維向量空間的子空間數。令 _q_ 表示一個有限域里的元素數目，則在 _q_ 元有限域上 _n_ 維向量空間的 _k_ 維子空間數等於

　
: ${\binom {n}{k}}_{q}.$

 

## 示例

　
: ${0 \choose 0}_{q}={1 \choose 0}_{q}=1$

　
: ${1 \choose 1}_{q}={\frac {1-q}{1-q}}=1$

　
: ${2 \choose 1}_{q}={\frac {1-q^{2}}{1-q}}=1+q$

　
: ${3 \choose 1}_{q}={\frac {1-q^{3}}{1-q}}=1+q+q^{2}$

　
: ${3 \choose 2}_{q}={\frac {(1-q^{3})(1-q^{2})}{(1-q)(1-q^{2})}}=1+q+q^{2}$

　
: ${4 \choose 2}_{q}={\frac {(1-q^{4})(1-q^{3})}{(1-q)(1-q^{2})}}=(1+q^{2})(1+q+q^{2})=1+q+2q^{2}+q^{3}+q^{4}$

## 性質

和普通二項式係數一樣，高斯二項式係數是中心對稱的:

　
: ${m \choose r}_{q}={m \choose m-r}_{q}.$

特別地，

　
: ${m \choose 0}_{q}={m \choose m}_{q}=1\,,$

　
: ${m \choose 1}_{q}={m \choose m-1}_{q}={\frac {1-q^{m}}{1-q}}=1+q+\cdots +q^{m-1}\quad m\geq 1\,.$

當 _q_ = 1 時，有

　
: ${m \choose r}_{1}={m \choose r}$

 

## 參考文獻

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