I'm trying to define $\frac{\mathrm d^m}{\mathrm dt^m}\mathbf P^n$ for a general matrix $\mathbf P$ (where each element is differentiated independently such that $(\dot{\mathbf P})_{ij} = \dot{(\mathbf P_{ij})}$). (I don't know why.)
Some examples just using the chain rule:
- $\dot{(\mathbf P^3)} = \dot{\mathbf P}\mathbf P^2+\mathbf P\dot{\mathbf P}\mathbf P+\mathbf P^2\dot{\mathbf P}$
- $\ddot{(\mathbf P^3)} = \ddot{\mathbf P}\mathbf P^2+\mathbf P\ddot{\mathbf P}\mathbf P+\mathbf P^2\ddot{\mathbf P}+\mathbf P\dot{\mathbf P}^2+\dot{\mathbf P}\mathbf P\dot{\mathbf P}+\dot{\mathbf P}^2\mathbf P$
Evidently there is an equivalence here between the derivatives and the weak $n$-compositions of $m$:
- $\dot{(\mathbf P^3)} \to (1,3) \to$ 100 010 001
- $\ddot{(\mathbf P^3)} \to (2,3) \to$ 200 020 002 011 101 110
So $$\frac{\mathrm d^m}{\mathrm dt^m}\mathbf P^n = \sum_{i=1}^{{m+n-1}\choose{n-1}}\left(\prod_{j=1}^{n}\frac{\mathrm d^f}{\mathrm dt^f}\mathbf P \right)$$ where $f$ is the $j$th part of the $i$th weak $n$-composition of $m$.
How can I define $f$?
What's a nice systematic ordering of the $i$ compositions such that $i$ is meaningful? (Above, I've just ordered them to make my TeX copypasting easier.)
I think a good way to write it is $$ \frac{d^m}{dt^m}\def\P{{\bf P}}\P^n=\sum_{\hspace{.7cm}\alpha\in \mathbb N^n\\\alpha_1+\dots+\alpha_n=m\\}\!\!\!\prod_{i=1}^n \frac{d^{\alpha_i}}{dt^{\alpha_i}}\P $$ Even better, just write it as $\sum_\alpha \prod_{i=1}^n (\frac{d}{dt})^{\alpha_i}\P$, and explain in words that $\alpha$ ranges over vectors of $n$ integers whose sum is $m$.
More in line with what you were asking, order the weak compositions lexicographically, define $\alpha_i$ to be the $i^{th}$ smallest weak composition in this order, and let $f=(\alpha_i)_j$. So, it would be $$ \sum_{i=1}^\binom{m+n-1}{n-1}\prod_{j=1}^n \left(\frac{d}{dt}\right)^{(\alpha_i)_j}\P $$