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 $

Reference

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