Generalized Vandermonde's identity Can you please provide a reference to the following generalization of Vandermonde's identity?
Given a positive integer $k$ and nonnegative integers $n_1, n_2, \ldots, n_k$ and $m$, it holds that $$\sum_{i_1+i_2+\cdots+i_k=m} \binom{n_1}{i_1} \binom{n_2}{i_2} \cdots \binom{n_k}{i_k} = \binom{n_1+n_2+\cdots+n_k}{m}.$$ The proof is well-known and based on the idea of counting in two different ways the coefficient of $x^m$ in the polynomial $(1+x)^{n_1+n_2+\cdots+n_k}$.

\begin{array}{l}
 \sum\limits_{i_1  + i_2  +  \cdots  + i_k  = m} {\left( \begin{array}{c}
 n_1  \\ 
 i_1  \\ 
 \end{array} \right)\left( \begin{array}{c}
 n_2  \\ 
 i_2  \\ 
 \end{array} \right)\left( \begin{array}{c}
 n_3  \\ 
 i_3  \\ 
 \end{array} \right) \cdots \left( \begin{array}{c}
 n_k  \\ 
 i_k  \\ 
 \end{array} \right)}  =  \\ 
  = \sum\limits_{i_1  + i_2  = m - \left( {i_3  \cdots  + i_k } \right)} {\left( {\left( \begin{array}{c}
 n_1  \\ 
 i_1  \\ 
 \end{array} \right)\left( \begin{array}{c}
 n_2  \\ 
 i_2  \\ 
 \end{array} \right)} \right)\left( \begin{array}{c}
 n_3  \\ 
 i_3  \\ 
 \end{array} \right) \cdots \left( \begin{array}{c}
 n_k  \\ 
 i_k  \\ 
 \end{array} \right)}  =  \\ 
  = \sum\limits_{i_3  = m - \left( {i_4  \cdots  + i_k } \right)} {\left( {\left( \begin{array}{c}
 n_1  + n_2  \\ 
 m - \left( {i_3  \cdots  + i_k } \right) \\ 
 \end{array} \right)} \right)\left( \begin{array}{c}
 n_3  \\ 
 i_3  \\ 
 \end{array} \right) \cdots \left( \begin{array}{c}
 n_k  \\ 
 i_k  \\ 
 \end{array} \right)}  =  \\ 
  =  \cdots  \\ 
 \end{array}
