If $\alpha \in\mathbb{N}^n$ then what is $\alpha!$?

Question

If $\alpha \in\mathbb{N}^n$ then what is ($\alpha!$) ?

Answer 1

It's simply the product of all the factorials: Suppose you have $\alpha=(\alpha_1,...,\alpha_n)\in{\mathbb{N}^n}$, then $$ \alpha!=\alpha_1!\cdot...\cdot\alpha_n! $$

Answer 2

By definition, if $\alpha:=(\alpha_1,\cdots,\alpha_n)\in\mathbb{N}^n$, one has: $$\alpha!=\alpha_1!\times\cdots\times\alpha_n!.$$ This is a convenient notation to write down Taylor's formula for multivariate functions.