# 范德蒙恒等式

${\displaystyle {\binom {n+m}{k}}=\sum _{i=0}^{k}{\binom {n}{i}}{\binom {m}{k-i}}}$

## 证明

### 母函数方法

${\displaystyle (1+x)^{n}(1+x)^{m}=(1+x)^{n+m}}$

${\displaystyle (1+x)^{n}(1+x)^{m}=\left(\sum _{i=0}^{n}{\binom {n}{i}}x^{i}\right)\left(\sum _{j=0}^{m}{\binom {m}{j}}x^{j}\right)=\sum _{k=0}^{m+n}\left(\sum _{i=0}^{k}{\binom {n}{i}}{\binom {m}{k-i}}\right)x^{k}}$

${\displaystyle (1+x)^{n+m}=\sum _{k=0}^{n+m}{\binom {n+m}{k}}x^{k}}$

${\displaystyle {\binom {n+m}{k}}=\sum _{i=0}^{k}{\binom {n}{i}}{\binom {m}{k-i}}}$ [1]

## 推广

### 多变量型

${\displaystyle \sum _{k_{ij}}{n_{1} \choose k_{11},k_{12},\dots ,k_{1t}}\dots {n_{s} \choose k_{s1},k_{s2},\dots ,k_{st}}={n_{1}+n_{2}+\dots +n_{s} \choose r_{1},r_{2},\dots ,r_{t}}}$

### 超几何函数

${\displaystyle {}_{2}F_{1}(a,b;c;1)=\sum _{n=0}^{\infty }{\frac {a^{(n)}b^{(n)}}{c^{(n)}n!}}={\frac {\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}},\quad \Re (c)>\Re (a+b)}$  [3]

${\displaystyle \sum _{i=0}^{k}{\binom {n}{i}}{\binom {m}{k-i}}={\frac {m!}{k!(m-k)!}}\sum _{i=0}^{\infty }{\frac {(-n)^{(i)}(-k)^{(i)}}{(m-k+1)^{(i)}i!}}={\frac {m!}{k!(m-k)!}}{}_{2}F_{1}(-n,-k;m-k+1;1)}$

${\displaystyle ={\frac {m!}{k!(m-k)!}}{\frac {\Gamma (m-k+1)\Gamma (n+m+1)}{\Gamma (n+m-k+1)\Gamma (m+1)}}={\frac {(n+m)!}{k!(n+m-k)!}}={\binom {n+m}{k}}}$

## 参考资料

1. 李松槐 杨伏香. 用数学模型证明范得蒙(Vandermonde)恒等式. 河南教育学院学报(自然科学版). 1999, (2) [2015-09-20]. （原始内容存档于2020-01-15）.
2. ^ Hac`ene Belbachir. A combinatorial contribution to the multinomial Chu-Vandermonde convolution (PDF). RECITS Laboratory. 2014 [2018-06-12]. （原始内容 (PDF)存档于2020-11-30）.
3. ^ Bailey, W.N. Generalized Hypergeometric Series (PDF). Cambridge University Press. 1935 [2018-06-12]. （原始内容 (PDF)存档于2017-06-24）.