广义二项式定理与生成函数

$$ \left( x+y \right) ^k=\sum_{n\ge 0}{\left( \begin{array}{c} k\\ n\\ \end{array} \right) x^ny^{k-n},\text{其中}k\text{为任意实数。}} \\ \text{其中}\left( \begin{array}{c} k\\ n\\ \end{array} \right) \text{定义为}\frac{k\left( k-1 \right) \left( k-2 \right) …\left( k-n+1 \right)}{n!} \\ \text{对于形如}\frac{1}{\left( 1-x \right) ^k}=\left( 1-x \right) ^{-k}\text{的生成函数，则有} \\ \left( 1-x \right) ^{-k}=\sum_{n\ge 0}{\left( \begin{array}{c} -k\\ n\\ \end{array} \right) \left( -x \right) ^n=\sum_{n\ge 0}{\left( -1 \right) ^n\left( \begin{array}{c} -k\\ n\\ \end{array} \right) x^n}} \\ \text{对于}\left( \begin{array}{c} -k\\ n\\ \end{array} \right) =\frac{\left( -k \right) \left( -k-1 \right) \left( -k-2 \right) …\left( -k-n+1 \right)}{n!}=\left( -1 \right) ^n\frac{k\left( k+1 \right) \left( k+2 \right) \left( k+3 \right) …\left( k+n-1 \right)}{n!}=\left( -1 \right) ^n\left( \begin{array}{c} k+n-1\\ n\\ \end{array} \right) \\ \text{故}\left( 1-x \right) ^{-k}=\sum_{n\ge 0}{\left( -1 \right) ^{2n}\left( \begin{array}{c} k+n-1\\ n\\ \end{array} \right) x^n}=\sum_{n\ge 0}{\left( \begin{array}{c} k+n-1\\ n\\ \end{array} \right) x^n} $$