Binomial Theorem/Multiindex

Theorem
Let $\alpha$ be a multiindex, indexed by $\left\{1,\ldots,n\right\}$ such that $\alpha_j \geq 0$ for $j = 1,\ldots,n$.

Let $x = (x_1,\ldots,x_n)$ and $y = (y_1,\ldots,y_n)$ be ordered tuples of real numbers.

Then:
 * $\displaystyle \left({x + y}\right)^\alpha = \sum_{0 \mathop \le \beta \mathop \le \alpha} {\alpha \choose \beta} x^{\beta} y^{\alpha-\beta}$

where $\displaystyle {n \choose k}$ is a binomial coefficient.

Proof
First of all, by definition of multiindexed powers we have:
 * $\displaystyle \left(x + y\right)^\alpha = \prod_{k \mathop = 1}^n\left(x_k + y_k\right)^{\alpha_k}$

We therefore calculate

On the other hand, we calculate:

This shows that:
 * $\displaystyle \left({x + y}\right)^\alpha = \sum_{0 \mathop \le \beta \mathop \le \alpha} {\alpha \choose \beta} x^{\beta} y^{\alpha-\beta}$

as required.