How to write the expression with multi-index notation?

How to write the expression with multi-index notation?

I need some help for writing the following expression with multi-index
notation, $$\sum_{i_1, \ldots, i_p=1}^n \frac{\partial^{2p}}{\partial
x_{i_1}^2\ldots \partial x_{i_p}^2}f(x, \xi),$$ where $f:\mathbb R^n\times
\mathbb T^n\rightarrow \mathbb C$. The above expression is simply the
Laplacian with respect to $x$ applied $p$ times. Any help will be
valuable. Thanks