"Transseries" are a kind of generalized power series that allow things like fractional exponents and exponentials (with another transseries as the exponent). I know very little about them but I have looked through e.g. Edgar's [Transseries for beginners](http://arxiv.org/abs/0801.4877). The transseries considered there are "expansions around infinity", so that the powers of the variable $x$ must *decrease* to the right; thus for instance the usual power series $\sum\_{n=0}^\infty \frac{x^n}{n!}$ for $e^x$ is not a transseries (at $\infty$), but we do I think have a transseries


$$ \sum\_{n=0}^\infty \frac{1}{n! x^n}$$


that is a sort of power series at $\infty$ for $e^{1/x}$.


On the other hand, unless I am confused, $e^{1/x}$ is also *itself* a transseries. So it seems that in the world of transseries we have two different ways to represent the same function $f(x)=e^{1/x}$, as "$e^{1/x}$" and as the above series "$ \sum\_{n=0}^\infty \frac{1}{n! x^n}$".


My question is, assuming this is correct, how should I think about this redundancy? Is it an unavoidable disadvantage of the extra expressivity of transseries over power series? Or is there some positive reason why we *want* to distinguish "$e^{1/x}$" and "$ \sum\_{n=0}^\infty \frac{1}{n! x^n}$"?



Redundancy in transseries representation of functions?
How embarassing; I puzzled over this for several days, only to figure out the answer an hour after posting the question.


A transseries is only allowed to contain exponentials $e^T$ when $T$ is a "purely large" transseries: one all of whose terms "go to infinity as $x$ does". That rules out $1/x$ as an allowable exponent. I hope/suppose that this restriction actually suffices to rule out any other apparent redundancies of this sort.


