A note on G-operators of order 2

TL;DR

This paper characterizes the form of G-function solutions for certain second-order differential equations, extending known results from first-order cases and encompassing broader classes like Nilsson-Gevrey series.

Contribution

It determines the structure of G-function solutions of inhomogeneous order 1 equations and second-order G-functions with algebraic dependence, advancing understanding beyond the first-order case.

Findings

Characterization of G-function solutions for inhomogeneous order 1 equations.

Description of G-functions of order 2 with algebraic dependence with their derivatives.

Extension of results to Nilsson-Gevrey arithmetic series.

Abstract

It is known that $G$-functions solutions of a linear differential equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we determine the form of a $G$-function solution of an inhomogeneous equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, as well as that of a $G$-function $f$ of differential order 2 over $\overline{\mathbb{Q}}(z)$, and such that $f$ and $f'$ are algebraically dependent over $\mathbb{C}(z)$. Our results apply more generally to Nilsson-Gevrey arithmetic series of order 0 that encompass $G$-functions.