Detailed generating function
Preliminaries
Generalized binomial theorem
Generalized binomial theorem is the general binomial theorem from the integer field to the field of real numbers
definition:
\[C_{\alpha}^{k}=\begin{cases} \frac{\alpha(\alpha-1)(\alpha-2) \dots (\alpha-k+1)}{k!},k>1 \\ 1,k=0 \\ 0,k<0 \end{cases}(k \in \mathbb{Z},\alpha \in \mathbb{R})\]
Then there
\[(x+y)^{\alpha}=\sum_{k=0}^{\infin} C_{\alpha}^{k} x^{\alpha-k}y^k (\alpha \in \mathbb{R})\]
inference:
(1) \[(x+y)^n=\sum_{k=0}^{n} C_{n}^{k} x^{n-k}y^k (n \in \mathbb{N^+})\]
Proof: split into two parts
\[ \begin{aligned} (x+y)^n &=\sum_{k=0}^{\infin} C_{n}^{k} x^{n-k}y^k \\ &=\sum_{k=0}^{n} C_{n}^{k} x^{n-k}y^k+ \sum_{k=n+1}^{\infin} C_{n}^{k} x^{n-k}y^k\end{aligned}\]
Note that when \ (n-, K \ in \ mathbb N \) , and \ (n <k \) when, \ (\ {n-FRAC (. 1-n-) (2-n-) \ DOTS (n-K +. 1- )} {k!} \) molecule there must be a 0, so \ (C_n ^ k = 0 \ )
Then \ ((x + y) ^ n = \ sum_ {k = 0} ^ {n} C_ {n} ^ {k} x ^ {nk} y ^ k (n \ in \ mathbb {N ^ +}) \)
(2) \[(x+y)^{-n}=\sum_{k=0}^{\infin} (-1)^k C_{n+k-1}^{n-1} x^{n-k}y^k (n \in \mathbb{N^+})\]
prove:
\[\begin{aligned} C_{-n}^{k} &=\frac{(-n)(-n-1)(-n-2) \dots (-n-k+1)}{k!} \\ &= (-1)^k \frac{n(n+1)(n+2) \dots (n+k-1)}{k!} \\ &=(-1)^k C_{n+k-1}^{k}=(-1)^k C_{n+k-1}^{n-1}\end{aligned}\]
Into Expression generalized binomial theorem,
\[(x+y)^{-n}=\sum_{k=0}^{\infin}(-1)^k C_{n+k-1}^{n-1} x^{n-k}y^k (n \in \mathbb{N^+})\]