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

2012/01/25 by Shweta Sharma, Sharma, Shweta
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods for differential equations #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1201.5266

openalex publication_date 2012/01/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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 ∈ ℤ.

Related

• Power series with sum-product Taylor coefficients and their resurgence algebra
• Arithmetic properties of the Taylor coefficients of differentially algebraic power series
• Spaces of functions satisfying simple differential equations
• Introduction to 1-summability and resurgence
• An arithmetic characterization of some algebraic functions and a new proof of an algebraicity prediction by Golyshev