“Vandermonde’s convolution formula”的概念、定义、翻译、参考文献-科学参考

Vandermonde’s convolution formula

Mathematics

The following relationship between binomial coefficients:

$(\begin{matrix} m + n \\ r \end{matrix}) = (\begin{matrix} m \\ 0 \end{matrix}) (\begin{matrix} n \\ r \end{matrix}) + (\begin{matrix} m \\ 1 \end{matrix}) (\begin{matrix} n \\ r - 1 \end{matrix}) + \dots + (\begin{matrix} m \\ r \end{matrix}) (\begin{matrix} n \\ 0 \end{matrix}) .$

The formula may be proved by equating the coefficients of x^r on both sides of the identity (1 + x)^m + n = (1 + x)^m(1 + x)ⁿ.

单词	Vandermonde’s convolution formula
释义	Vandermonde’s convolution formula Mathematics The following relationship between binomial coefficients: $(\begin{matrix} m + n \\ r \end{matrix}) = (\begin{matrix} m \\ 0 \end{matrix}) (\begin{matrix} n \\ r \end{matrix}) + (\begin{matrix} m \\ 1 \end{matrix}) (\begin{matrix} n \\ r - 1 \end{matrix}) + \dots + (\begin{matrix} m \\ r \end{matrix}) (\begin{matrix} n \\ 0 \end{matrix}) .$ The formula may be proved by equating the coefficients of x^r on both sides of the identity (1 + x)^m + n = (1 + x)^m(1 + x)ⁿ.
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