A class of arithmetic difference operators

We define and study a new class of difference operators in $\overline{\mathbb Q}[n,\sigma]$, called $\varSigma$-operators. By analogy with the definition of $E$-operators by Andr\'e, $\varSigma$-operators are images of $G$-operators in $\overline{\mathbb Q}[z,\delta]$ by the morphism of $\mathbb C$-algebras defined by $z\to \sigma$ and $\delta \mapsto n$. Here $\delta=z\frac{d}{dz}$ is Euler's operator and $\sigma$ is the backward shift operator. We prove various results about these difference operators among which: determination of their slopes and generalized exponents, purity and asymptotic behavior of their solutions. These properties have a strong arithmetic coloration, in relation with the theory of $G$-functions. We illustrate our results by revisiting various classical and more recent examples.