Power series with Taylor coefficients of sum-product type and algebraic differential equations /SP-series and the interplay between their resurgence and differential properties

TL;DR

This paper explores the differential properties of sum-product series (SP-series) and their singularities, revealing that their inner generators satisfy linear ODEs with polynomial coefficients for certain inputs, but likely not for rational, non-monomial functions.

Contribution

It provides explicit calculations and numerical evidence on the differential equations satisfied by inner generators of SP-series, highlighting differences based on input types.

Findings

Inner generators for polynomial and monomial inputs satisfy linear ODEs.

Numerical evidence suggests non-existence of ODEs for certain rational, non-monomial inputs.

Explicit calculations support the theoretical framework of SP-series and their resurgence properties.

Abstract

The article is based on the differential properties of the inner generators (singularities) that occur while handling SP series (sum product series), power series whose Taylor coefficients can be written as sum-product combinations. It is an elaboration with numerical details and some explicit calculations of the chapter 6 of the joint article with Ecalle on these series. For all polynomial inputs $f$ and all monomial inputs $F$, the inner generators corresponding to them verify ordinary differential equations of linear homogeneous type with polynomial coefficients. Numerical results strongly suggest the non-existence of ODEs for functions $F$ that are rational but not monomial, i.e. not of the form ${(1-ax)}^p$, for $p \in \mathbb{Z}$.