MultiIndex

A multi-index is simply a tuple of natural numbers, for which the following operations are defined. For two n-dimensional multi-indices $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)\in\mathbb{N}_0^n$ $\beta=(\beta_1,\beta_2,\ldots,\beta_n)\in\mathbb{N}_0^n$ , we define

Partial ordering

$\beta\geq\alpha\Leftrightarrow\beta_i\geq\alpha_i\;\forall\;i\in\mathbb{N}^n\;,$

Component-wise summation

$\alpha\pm\beta=\left(\alpha_1\pm\beta_1,\alpha_2\pm\beta_2,\ldots,\alpha_n\pm\beta_n\right)\;,$

Absolute value

$\left|\alpha\right|=\sum_{i=1}^n\alpha_i\;,$

Factorial

$\alpha!=\prod_{i=1}^n\alpha_i!\;,$

Power (for $\b{x}\in\mathbb{R}^n\.)$

$\b{x}^\alpha=\prod_{i=1}^nx_i\alpha_i\;.$

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