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In article <wada-1103961143170001@tao.ee.ibaraki.ac.jp>
wada@ee.ibaraki.ac.jp (Tatsuaki Wada) writes:
> $B!!(BA$B$H(BB$B$,Hs2D49$N$H$-!"(B
>       Exp[A] Exp[B] = Exp[C]
> $B$H$9$k$H!"(BC$B$r(BA$B$H(BB$B$GI=$o$9!#(B

 (Baker-)Campbell-Hausdorff $B$N8x<0$H$7$FCN$i$l$F$*$j(B,
$B4dGH$N(B$B?t3X<-E5(B ($BBh(B 3$BHG(B) $B$N(B
 425$B%j!<72(B R. $BBh0lI8=`:BI87O$K$*$1$k>hK!4X?t(B
$B$N9`$K$b:\$C$F$$$^$9(B.
 ($B$7$+$7IaDL$O(B$B0z$1$J$$$G$7$g$&$M(B.)

C = c_1(A, B) + c_2(A, B) + \cdots,
c_1(A, B) = A + B,
(n+1)c_{n+1}(A, B)
 = (1/2)[A - B, c_n(A, B)]
   + \sum_{1 \leq p, 2p \leq n} K_{2p} \sum_{(k_1, \ldots, k_{2p}) \in A_n}
       [c_{k_1}(A, B), [ \cdots [c_{k_{2p}}(A, B), A + B] \cdots ]],

$B$H$$$&A22=<0$,M?$($i$l$F$*$j$^$9(B. $BC"$7(B,

{z \over 1 - e^{-z}} - (1/2) z (= (z/2)coth(z/2))
 = 1 + \sum_{p=1}^\infty K_{2p} z^{2p},
A_n = \{ (k_1, \ldots, k_{2p}) \in N^{2p} | \sum_{i=1}^{2p} k_i = n \}.

K_2, K_4, \dots $B$O(B Bernoulli$B?t$r(B$B;H$C$F=q$1$^$9(B.

2 c_2(A, B) = (1/2)[A - B, A + B] = [A, B],
3 c_3(A, B)
 = (1/2)[A - B, (1/2)[A, B]]
   + K_2 [A + B, [A + B, A + B]]
 = (1/4)[A, [A, B]] + (1/4)[B, [B, A]]
 = (1/4)[[B, A], A] + (1/4)[[A, B], B].

> $B9b669/$5$s$N!V8x<0=8!!NL;RNO3X$H$=$N<~JU!W!!!J9VCL<R%5%$%(%s%F%#%U%#%/!K(B
> $B$N(Bp103$B$K$O!"(B
> 
> $B!!!!!!(BC = A + B + 1/2 [A, B] + 1/12 [[A, B], B] + 1/12 [[B, A], A]
>             + 1/24[[[A, B], B, A] + ....
                    [[[A, B], B], A] (?)
> 
> $B$H$"$j$^$9$,!"(B

K_2 = 1/12,
4 c_4(A, B)
 = (1/2)[A - B, (1/12)[[A, B], B] + (1/12)[[B, A], A]]
   + K_2 [A + B, [(1/2)[A, B], A + B]]
 = (1/12)[A, [[A, B], B]] - (1/12)[B, [[B, A], A]]
 = - (1/12)[A, [B, [A, B]]] - (1/12)[B, [A, [A, B]]]
 = - (1/12)[[[A, B], B], A] - (1/12)[[[A, B], A], B].

[[A, B], C] = [[A, C], B] + [A, [B, C]],
[[A, B], [A, B]] = 0 $B$rMQ$$$F:G8e$N<0$rJQ7A$9$k$H(B

4 c_4(A, B)
 = - (1/6)[[[A, B], B], A].

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--
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Tsukamoto, C. : chiaki@ipc.kit.ac.jp