查看“︁高斯二项式系数”︁的源代码

{{expand|time=2013-10-23T03:43:40+00:00}}
'''高斯二项式系数''' (也称作 '''高斯系数''', '''高斯多项式''', 或 '''''q''-二项式系数''')在[[数学]]里是指[[二项式系数]]的[[q-模拟]]。

==定义==

高斯二项式系数被定义为：

<math>{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 \le m \\
0 & r>m \end{cases}</math>

其中， ''m'' 和 ''r'' 是非负整数。 当 {{nowrap|''r'' {{=}} 0}}时值为1。

高斯二项式系数计算一个有限维[[向量空间]]的子空间数。令''q''表示一个[[有限域]]里的元素数目，则在''q''元有限域上''n''维向量空间的''k''维子空间数等于
:<math>
\binom nk_q.
</math>


==示例==

:<math>{0 \choose 0}_q = {1 \choose 0}_q = 1</math>

:<math>{1 \choose 1}_q = \frac{1-q}{1-q}=1</math>

:<math>{2 \choose 1}_q = \frac{1-q^2}{1-q}=1+q</math>

:<math>{3 \choose 1}_q = \frac{1-q^3}{1-q}=1+q+q^2</math>

:<math>{3 \choose 2}_q = \frac{(1-q^3)(1-q^2)}{(1-q)(1-q^2)}=1+q+q^2</math>

:<math>{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</math>

==性质==
和普通二项式系数一样, 高斯二项式系数是中心对称的:

:<math>{m \choose r}_q = {m \choose m-r}_q. </math>

特别地,

:<math>{m \choose 0}_q ={m \choose m}_q=1 \, ,</math>

:<math>{m \choose 1}_q ={m \choose m-1}_q=\frac{1-q^m}{1-q}=1+q+ \cdots + q^{m-1} \quad m \ge 1 \, .</math>

当 {{nowrap|''q'' {{=}} 1}} 时，有

:<math>{m \choose r}_1 = {m \choose r}</math>


==参考文献==
*Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York:  Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914,  ISBN 0470274530, ISBN 978-0470274538

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[[Category:Q-analogs]]
[[Category:Factorial and binomial topics]]