Abstract

A differentially recursive sequence over a differential field is a sequence of elements satisfying a homogeneous differential equation with non-constant coefficients (namely, Taylor expansions of elements of the field) in the differential algebra of Hurwitz series. The main aim of this paper is to explore the space of all differentially recursive sequences over a given field with a non-zero differential. We show that these sequences form a two-sided vector space that admits, in a canonical way, a structure of Hopf algebroid over the subfield of constant elements. We prove that it is the direct limit, as a left comodule, of all spaces of formal solutions of linear differential equations and that it satisfies, as Hopf algebroid, an additional universal property. When the differential on the base field is zero, we recover the Hopf algebra structure of linearly recursive sequences.

摘要

微分场上的微分递归序列是满足Hurwitz级数微分代数中具有非常数系数（即，场元素的泰勒展开式）的齐次微分方程的元素序列。本文的主要目的是探索给定域上具有非零差分的所有差分递归序列的空间。我们证明了这些序列形成了一个双向向量空间，该空间以规范的方式允许常数元素子域上的Hopf代数结构。我们证明它是线性微分方程形式解的所有空间的直接极限，作为左协模，并且满足作为Hopf代数的附加泛型性质。当基础字段的差分为零时，