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

It is known that $G$-functions solving a linear differential equation of order $1$ with coefficients in $\overline {\mathbb Q}(z)$ are algebraic (and 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 solving 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 holonomic Nilsson–Gevrey arithmetic series of order 0 that encompass $G$-functions.