Multi-index notation ($\alpha =({\alpha }_1,\dots ,{\alpha }_n)$)

A multi-index is a tuple of nonnegative integers used to abbreviate products and partial derivatives: $|\alpha |={\sum }_j{\alpha }_j$, $x^{\alpha }={\prod }_j{x}_j^{{\alpha }_j}$, and $D^{\alpha }={\prod }_j{D}_j^{{\alpha }_j}$.