Why is it customary to have formal series at infinity in the context of resurgence and 1-summability?

Question

In the context of the theory of 1-summability and resurgence, it is customary to deal with formal series "at infinity" rather than at $0$ $$ \sum_{n=0}^{\infty}a_nz^{-n} $$ This is stated for example at the beginning of section 3 on https://arxiv.org/pdf/1405.0356.pdf. Why is this? is there really any preference over using $t=1/z$? $$ \sum_{n=0}^{\infty}a_nt^{n} $$

Answer 1

The tradition probably goes back to classical results on Laplace transform: Lalace transform of a function $f$ on the positive ray is $$Lf(\zeta)=\int_0^\infty f(z)e^{-\zeta z}dz.$$ If $f$ is of exponential type on the positive ray, this is holomorphic in $\Re\zeta>a$. Now if $f$ is entire, of exponential type, you can analytically continue this Laplace transform by rotating the integration path. This defines the Laplace transform in a full neighborhood of $\infty$, and it extends holomorphically to infinity and equals to $0$ these, that is $$F(\zeta)=\sum_{n=1}^\infty a_nz^{-n}.$$ There is a one-to one correspondence between entire functions $f$ of exponential type, and Functions $F$ analytic at $\infty$, such that $F(\infty)=0$. (Polya's theorem). Modern summation theory is a generalization of this in some sense, and it considers divergent series $F$. They prefer to keep notation consistent with the classical theory.