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 $\b{x}\in\mathbb{R}^n\.)$
• $\b{x}^\alpha=\prod_{i=1}^nx_i\alpha_i\;.$