The Baker-Campbell-Hausdorff problem is to obtain $\log(e^{X}e^{Y})$ where $X,Y$ are appropriate operators. The Dynkin series $$\log(e^{tX}e^{tY})=t(X+Y)+\frac{t^2}{2}[X,Y]+o(t^3)$$ gives an expansion in powers of a perturbation parameter $t$, with higher coefficients expressed in terms of successively more complicated commutators of $X,Y$. The parameter $t$ has been introduced to control the convergence of the series; so long as the operators $X,Y$ are bounded the Dynkin series has a finite radius of convergence (theorem 5 of Suzuki 1997). A more involved approach would be to replace $e^{t X}\to e^{s X}$; the appropriate Dynkin series would then presumably be in powers of $s,t$ and have finite radius of convergence in each variable.


Now suppose we instead consider the product $e^X e^{tY}$ for $t\gg 1$. My intuition had been that there would be an asymptotic series $$\log(e^{X} e^{tY})=tY+X+o(t^{-1}).$$ But this is clearly not the standard Dynkin series: powers of $t$ now only arise from the operator $Y$, and moreover $t=\infty$ will always fall outside the finite radius of convergence. Hence my question:



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> Is there a known asymptotic series for $\log(e^Xe^{t Y})$ at large $t$?
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References:
 - Suzuki, M. Commun.Math. Phys. (1977) 57: 193. doi:10.1007/BF01614161. 



Behavior of a Baker-Campbell-Hausdorff problem at infinity
It may be helpful to consider an example that can be solved exactly; Using the [Special-case closed form of the Baker-Campbell-Hausdorff formula](https://arxiv.org/abs/1501.02506) one finds that if the commutator $[X,Y]$ evaluates to
$$[X,Y]=uX +vY +cI$$
then the desired logarithm of the product of matrix exponentials equals
$$\log(e^X e^{tY})=tY+X+f(u,v,t)(uX +vY +cI)$$
$$f(u,v,t)=\frac{(ut-v)e^{ut+v}-ute^{ut}+ve^{v}}{uv(e^{ut}-e^{v})}$$
The large-$t$ limit can now be read off once the sign of $u$ is known:
$$\lim\_{t\rightarrow\infty}f(u,v,t)=\begin{cases}
-1/u&\text{if}\;u<0\\
(t/v)(e^v-1)&\text{if}\;u>0\\
(t/v)[v-1+v(e^v-1)^{-1}]&\text{if}\;u=0
\end{cases}$$
There are no terms greater than order $t$ in the large-$t$ limit for this class of commutators. *(I'm actually a bit puzzled how $t^2$ terms and higher might appear at all, an explicit example would help me a lot.)*


