## Multi-index power

### Set

 context $G$ … group context $g \in \text{Sequence}(G)$ context $\alpha \in \text{Sequence}(\mathbb N)$ context $\mathrm{length}(g)=\mathrm{length}(\alpha)$
 definiendum $\langle g,\alpha\rangle \mapsto g^\alpha := \prod_{i=1}^{\mathrm{length}(\alpha)} g_i^{\alpha_i}$

We also write $|\gamma|=\sum_i^{\mathrm{length}(\gamma)} \gamma_i$.

### Discussion

In most cases, the base sequence is understood. E.g. if

$\gamma=\langle 3,1,0,0,2 \rangle$

is taken to be a multiindex, then $|\gamma|=6$ and we write

$f^{(\gamma)}(x) \equiv \frac{\partial^{|\gamma|}}{\partial x^\gamma} f \equiv \frac{\partial^3}{\partial x_1^3} \frac{\partial}{\partial x_2} \frac{\partial^2}{\partial x_5^2} f$