Simple multi-index question

[ajb]

First, I have never really liked multi-index notation.

 

I wish to find a neat expression for

 

$\partial_{\mathbf{A}}(f(x))^{l} = \frac{\partial^{|\mathbf{A}|}}{\partial x^{A_{1}}\cdots \partial x^{A_{n}}} (f(x))^{l}$.

 

Here $\mathbf{A}$ is a multi-index, but $l$ is an exponent.

 

One should be able to you the Leibniz rule recursively to get a nice expression. (But note that $\partial_{\mathbf{A}}$ is not a derivation).

 

Has anyone seen the answer to this anywhere? I am sure it has been calculated before now. Can anyone save me the time and effort in doing it myself? I expect it is in a book on pseudo-differential operators. (I can't get to the library today).

 

Thanks

 

It does not matter to much now, as I have found exactly what I need in paper I am reading.

 

Still, thanks to those people who read this post.