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 $(1+x)^{n_1+n_2+\cdots+n_k}$.

  • $\begingroup$ Here you can find a formulation and a sketch of a proof. $\endgroup$ – Pp.. Feb 3 '15 at 5:09
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    $\begingroup$ @Pp. Your link points to this thread. In any case, I don't need a proof, but a (standard) reference to the result (say, a book or an article published in a mainstream journal). $\endgroup$ – Grinch Feb 3 '15 at 5:21
  • $\begingroup$ I would try searching in A=B or Concrete Mathematics. $\endgroup$ – Pp.. Feb 3 '15 at 5:25
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    $\begingroup$ As far as I can say, it is neither there, nor in Riordan's book, nor in Gould's tables, nor in Stanley's two volumes. $\endgroup$ – Grinch Feb 3 '15 at 5:28

The generalised identity is stated without proof in Concrete Mathematics, exercise 5.62. The proof technique that you mention is used for the ungeneralised Vandermonde identity in section 5.4.

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