Baker-Campell-Hausdorff

• To: mathgroup at smc.vnet.net
• Subject: [mg58071] Baker-Campell-Hausdorff
• From: "Shug Boabby" <Shug.Boabby at gmail.com>
• Date: Fri, 17 Jun 2005 05:20:28 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

hi there,

does anyone know of a function to return the Baker-Campell-Hausdorff
formula to an certain order?

see
http://planetmath.org/encyclopedia/BakerCampellHausdorffFormulae.html
for a brief and over-simplified description... i'm afraid the actual
recursive equation only exists in Lie algebra textbooks, though, for
the LaTeX inclined, this should print the actual recursive generating
function (you may need to \usepackage{amsmath,amsfonts,amssymb}):

%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\label{eq:BCH}
(n+1)\mathrm{C}_{n+1}(X:Y)&&=\frac 12 \cb{X-Y}{\mathrm{C}_n(X:Y)}
\\ \nonumber + \sum_{p \ge 1, 2p \le n} K_{2p}\!\!\!\!\!
\sum_{\substack{k_1,\ldots,k_{2p}> 0 \\ k_1+\ldots+k_{2p}=n }}&&
\!\!\!\!\!\!\!\!\!\!\cb{\mathrm{C}_{k_1}(X:Y)}
{\left[\ldots,\cb{\mathrm{C}_{k_{2p}}(X:Y)}{X+Y}\ldots\right]}
\end{eqnarray}
where ($n \ge 1 ; X,Y\in \mathfrak g$) and $\mathrm{C}_1(X:Y)=X+Y$. The
$K$s are defined by
$$\label{eq:BCH:K} \frac{z}{1-e^{-z}}-\frac z2-1=\sum_{p=1}K_{2p}z^{2p}$$
which are closely related to the Bernoulli numbers ($B_n$), defined by
$$\label{eq:BCH:bernoulli} \sum_{n=0}\frac{B_nx^n}{n!}=\frac{x}{e^x-1}$$
%%%%%%%%%%%%%%%%%%%%

i have implemented this in [another CAS] with the help of an external C
program (to do the partition), but i am keen to re-implement all my
current work in Mathematica and a lot of that work requires this BCH
formula.

the function myself and post it here if thats ok? i've noticed 3 or 4
requests over the last decade for such a function, but no replies; so
it might help someone one day.

incidentally, since this all requires non-commutative multiplication, i
was wondering if anyone knows of a Commutator implementation? the docs
have me stumped and there appears to be no Commutator function/notation
in mathematica, though this helpful document exists
http://mathworld.wolfram.com/Commutator.html
the apparent lack of a commutator bracket package is quite troubling.
though i realise the dot . operator is for non-commutative
multiplication. having all the Jacobi identities in the simplification
routines would be handy.



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