Multi-index

In mathematics, multi-index is an n-tuple of non-negative integers. Multi-indices are widely used in multidimensional analysis to denote e.g. partial derivatives and multivariable power function. Many formulas known from the one dimension one (i.e. the real line) carry on to $\mathbb {R} ^{n}$ by simple replacing usual indices with multi-indices.

Formally, multi-index $\alpha$ is defined as

$\alpha =(\alpha _{1},\,\alpha _{1},\,\ldots ,\alpha _{n})$, where $\alpha _{i}\in \mathbb {N} \cup \{0\}.$

Basic definitions and notational conventions using multi-indices.

• the order or length of $\alpha$

$|\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}$

• factorial

$\alpha !=\alpha _{1}!\cdot \alpha _{2}!\cdots \alpha _{n}!$