The Ring of Factorially Divergent Power Series

Abstract

The content of this chapter, which is based on the author’s article (Borinsky, Generating asymptotics for factorially divergent sequences, 2016, [1]), is solely concerned with sequences $f_n$, which admit an asymptotic expansion for large n of the form,

$$\begin{aligned} f_n = \alpha ^{n+\beta } \Gamma (n+\beta ) \left( c_0 + \frac{c_1}{\alpha (n+\beta -1)} + \frac{c_2}{\alpha ^2(n+\beta -1)(n+\beta -2)} + \cdots \right) , \end{aligned}$$

for some $\alpha \in \mathbb {R}_{>0}$, $\beta \in \mathbb {R}$ and $c_k \in \mathbb {R}$ as they appeared in the statement of Theorem 3.3.1. The theory of these sequences is independent of the theory of zero-dimensional QFT and graphical enumeration, but necessary to analyze the asymptotics for these problems.