differential operator power coefficients - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T14:18:08Z http://mathoverflow.net/feeds/question/80828 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80828/differential-operator-power-coefficients differential operator power coefficients zroslav 2011-11-13T16:58:56Z 2011-11-14T19:07:21Z <p>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?</p> http://mathoverflow.net/questions/80828/differential-operator-power-coefficients/80873#80873 Answer by Gjergji Zaimi for differential operator power coefficients Gjergji Zaimi 2011-11-14T07:08:18Z 2011-11-14T19:07:21Z <p>Let me start by giving the formula</p> <p>$$H_{n,l}(y_0,y_1,\dots,y_n)=\sum_{(k_1,k_2,\dots,k_{n-1})\in P_{n,l}}\frac{y_{0}}{l!}\prod_{j=1}^n (j+1-k_1-\cdots-k_j)\frac{y_{k_j}}{k_j!}$$</p> <p>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 $1\le i \le n$. </p> <p>In this form this is due to L. Comtet: </p> <blockquote> <p>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</p> </blockquote> <p>This has an <a href="http://oeis.org/A139605" rel="nofollow">OEIS entry</a>, where you can find more references. In particular see Bergeron, F. and Reutenauer, C., Une interpretation combinatoire des puissances d'un operateur differentiel lineaire, Ann. Sci. Math. Quebec. 11, 269-278 (1987) for a combinatorial interpretation in terms of forests of rooted trees.</p> <p>The analogous expansion for the multivariable case is treated in <a href="http://www.springerlink.com/content/g58q46214245p42m/" rel="nofollow">"Universal expansion of the powers of a derivation"</a> by M. Ginocchio. There are also generalizations to the non-commutative case that are described in his other article <a href="http://www.sciencedirect.com/science/article/pii/S0012365X97000812" rel="nofollow">"On the Hopf algebra of functional graphs and differential algebras"</a>. </p>