Let $(F(x)\frac{d}{dx})^n=\sum_{i=1}^n H_{n,i}(F, F', F^{(2)}, \ldots , F^{(n)})\frac{d^i}{dx^i}$. I'm curious about the exact formula for $H_{n,i}(y_0, y_1, \ldots , y_n)$. What is known about it?
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1$\begingroup$ Are you assuming F(x) are polynomials? Also, where did you see this expression, it would help direct me to what you are looking for. If you don't assume polys, I'd expect the commutator to suck pretty bad if you take the Taylor polynomial approach then assume all of F's derivs exist(reasonable in this setting) but then idk how to interpret the infinite sums in the commutator. $\endgroup$– B. BischofCommented Nov 14, 2011 at 5:04
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1$\begingroup$ @Bischof: No, $F(x)$ is assumed to be an arbitrary function. This expression was seen by me in my mind: for example $(F(x)d/dx)^2=F^2d^2/dx^2+FF'd/dx$. The general case can be proved by induction by $n$ $\endgroup$– zroslavCommented Nov 14, 2011 at 7:40
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1$\begingroup$ I see now what you meant, I was thinking about this in the wrong way. $\endgroup$– B. BischofCommented Nov 14, 2011 at 9:12
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1 Answer
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Let me start by giving the formula
$$H_{n,l}(y_0,y_1,\dots,y_n)=\sum_{(k_1,k_2,\dots,k_{n-1})\in P_{n,l}}\frac{y_{0}}{l!}\prod_{j=1}^n (j+1-k_1-\cdots-k_j)\frac{y_{k_j}}{k_j!}$$
where, $P_{n.l}$ is the set of $(k_1,k_2,\dots,k_{n-1})\in \mathbb Z_{\geq 0}^{n-1}$ which satisfy $k_1+\cdots+k_{n-1}=n-l$ and $k_1+\cdots+k_i\le i$ for all $1\le i \le n$.
In this form this is due to L. Comtet:
L. Comtet, Une formule explicite pour les puissances successives de l'opérateur de dérivation de Lie, C. R. Acad. Sci. Paris Sér. A-B 276 (1973) A165–A168. Zbl 0252.05002
This has an OEIS entry, A139605, where you can find more references. In particular see Bergeron, F. and Reutenauer, C., Une interprétation combinatoire des puissances d'un opérateur différentiel linéaire, Ann. Sci. Math. Quebec. 11, 269–278 (1987) for a combinatorial interpretation in terms of forests of rooted trees.
The analogous expansion for the multivariable case is treated in "Universal expansion of the powers of a derivation" by M. Ginocchio. There are also generalizations to the non-commutative case that are described in his other article "On the Hopf algebra of functional graphs and differential algebras".
answered Nov 14, 2011 at 7:08
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1$\begingroup$ Don't you know about the expression $(\sum_{i=1}^n F_i(x)\frac{d}{dx_i})^n$ in terms of mixed partial derivatives of $F_i$? $\endgroup$– zroslavCommented Nov 14, 2011 at 8:01
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1$\begingroup$ @zroslav, I updated the post with a few more links to some recent papers. $\endgroup$ Commented Nov 14, 2011 at 8:42
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1$\begingroup$ See also "Recent Developments in Combinatorial Aspects of Normal Ordering" by Matthias Schork (2021) ecajournal.haifa.ac.il/Volume2021/ECA2021_S2B2.pdf $\endgroup$ Commented Feb 14, 2023 at 17:20