"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. 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}$"?

  • $\begingroup$ @GerryMyerson As I said, a transseries is allowed to contain exponentials with transseries exponent, as well as powers of x. The basic "monomial" appearing in a transseries is of the form $x^n e^T$, where $T$ is another transseries. $\endgroup$ – Mike Shulman Jan 11 '16 at 4:51
  • 1
    $\begingroup$ Oops! $T$ is not just any transseries, it has to be "purely large", which rules out $1/x$. $\endgroup$ – Mike Shulman Jan 11 '16 at 5:00
  • $\begingroup$ You can know the rules which allow to avoid redundancies by combining "Orders of infinity" (in the sense of Hardy) and Neumann-Malcev series. $\endgroup$ – Duchamp Gérard H. E. Nov 28 '17 at 5:46

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.