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I have seen the following mathematical identity in a book:

$$ \prod_{i=1}^{N}\left( 1 + ax_i \right)^c = \left( 1 + \sum_{i=1}^{N}{ax_i} + \cdots + a^N\prod_{i=1}^{N}{x_i} \right)^c $$

Is this a generalization of the Binomial theorem? What are the implicit terms within $\cdots$? Any explanations, proofs, references are appreciated!

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    $\begingroup$ Should that $a^N$ be on the outside of the product? For $c=1$ and $x_i=1$ you have (1+a)^N = 1 + \cdots + a^(N^2)$. $\endgroup$ – muaddib Jun 19 '15 at 12:16
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This follows from the "Multi-Binomial Theorem". Quoted from the wiki page for the Binomial Theorem

$$(x_{1}+y_{1})^{n_{1}}\dotsm(x_{d}+y_{d})^{n_{d}} = \sum_{k_{1}=0}^{n_{1}}\dotsm\sum_{k_{d}=0}^{n_{d}} \binom{n_{1}}{k_{1}}\, x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}\;\dotsc\;\binom{n_{d}}{k_{d}}\, x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}$$

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These are simply Vieta's formulas, expressing the relations between roots and coefficients.

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It is not a generalisation of the Binomial theorem because the exponent of $c$ isn't really handled -- they just took it outside. If you were to expand out the right-hand-side, you would have a generalisation of the Binomial theorem.

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