For a sequence x ∈ l 10, one can consider the achievement set E(x) of all subsums of series Σn=1∞ x(n). It is known that E(x) has one of the following structures: a finite union of closed intervals, a set homeomorphic to the Cantor set, a set homeomorphic to the set T of subsums of Σn=1∞ x(n) where c(2n − 1) = 3/4n and c(2n) = 2/4n (Cantorval). Based on ideas of Jones and Velleman [Jones R., Achievement sets of sequences, Amer. Math. Monthly, 2011, 118(6), 508–521] and Guthrie and Nymann [Guthrie...

Nous introduisons une version $q$-analogue du procédé d’accélération élémentaire d’Écalle-Martinet-Ramis et définissons la notion de série entière $Gq$-multisommable. Nous montrons que toute série entière solution formelle d’une équation aux $q$-différences linéaire analytique est $Gq$-multisommable.

This paper concerns difference equations $y(x+1)=G(x,y)$ where $G$ takes values in ${\mathbf{C}}^n$ and $G$ is meromorphic in $x$ in a neighborhood of $\infty$ in $\mathbf{C}$ and holomorphic in a neighborhood of 0 in ${\mathbf{C}}^n$. It is shown that under certain conditions on the linear part of $G$, formal power series solutions in $x^{-1/p},p\in \mathbf{N},$ are multisummable. Moreover, it is shown that formal solutions may always be lifted to holomorphic solutions in upper and lower halfplanes, but in general these solutions are not uniquely determined by the formal solutions.

In this paper a proof is given of a theorem of J. Écalle that formal power series solutions of nonlinear meromorphic differential equations are multisummable.