A note on $G$-operators of order $2$

It is known that G-functions solutions of a linear differential equation of order 1 with coefficients in 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 Q(z), as well as that of a G-function f of differential order 2 over Q(z) and such that f and f ′ are algebraically dependent over C(z). Our results apply more generally to holonomic Nilsson-Gevrey arithmetic series of order 0 that encompass G-functions.