Definition:Factorial/Multiindices

Definition
Let $\alpha$ be a multiindex, indexed by a set $J$ such that for each $j \in J$, $\alpha_j \ge 0$.

Then we define:
 * $\ds \alpha! = \prod_{j \mathop \in J} \alpha_j!$

where the factorial on the right is a factorial of natural numbers.

Note that by definition, all by finitely many of the $\alpha_j$ are zero, so the product over $J$ is convergent.