Using the techniques developed by Jean Ecalle for the study of nonlinear differential equations, we prove that the $q$-difference equation$$x{\sigma }_{q}y=y+b(y,x)$$with $\left({\sigma }_{q}f\right)\left(x\right)=f\left(qx\right)$ ($q\>1$) and $b(0,0)={\partial }_{y}b(0,0)=0$ is analytically conjugated to one of the following equations :$$x{\sigma }_{q}y=y\text{ou}x{\sigma }_{q}y=y+x$$

In this paper we introduce a new invariant for extensions of difference fields, the distant degree, and discuss its properties.

In ultrametric Nevanlinna theory, the Nevanlinna’s second main theorem for small functions has only been proved in the case of at most three small functions. In this paper, we prove a second main theorem for $q$ small functions of a special type when the residue characteristic of the field is zero.

Un algorithme est présenté pour calculer en toute généralité le « réseau de Levelt » ${\Lambda }_L$ pour un réseau $\Lambda$ donné.

We generalize and unify the proofs of several results on algebraic independence of arithmetic functions and Dirichlet series by using a theorem of Ax on the differential Schanuel conjecture. Along the way, we find counter-examples to some results in the literature.

We survey recent work on arithmetic analogues of ordinary and partial differential equations.