Multi-index

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

Formally, multi-index ${\displaystyle \alpha }$ is defined as

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

Basic definitions and notational conventions using multi-indices.

• The order or length of ${\displaystyle \alpha }$
${\displaystyle |\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}}$
• Factorial of a multi-index
${\displaystyle \alpha !=\alpha _{1}!\cdot \alpha _{2}!\cdots \alpha _{n}!}$
• multidimensional power notation
If ${\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})\in \mathbb {R} ^{n}}$ and ${\displaystyle \alpha =(\alpha _{1},\,\alpha _{1},\,\ldots ,\alpha _{n})}$ is a multi-index then ${\displaystyle x^{\alpha }}$ is defined as
${\displaystyle x^{\alpha }=(x_{1}^{\alpha _{1}},x_{2}^{\alpha _{2}},\ldots ,x_{n}^{\alpha _{n}})}$
• The following notation is used for partial derivatives of a function ${\displaystyle f:\mathbb {R} ^{n}\mapsto \mathbb {R} }$
${\displaystyle D^{\alpha }f={\frac {\partial ^{|\alpha |}f}{\partial x_{1}^{\alpha _{1}}\partial x_{2}^{\alpha _{2}}\cdots \partial x_{n}^{\alpha _{n}}}}}$
Remark: sometimes the symbol ${\displaystyle \partial ^{\alpha }}$ instead of ${\displaystyle D^{\alpha }}$ is used.