Accelero-summation of the formal solutions of nonlinear difference equations
The author studies the following nonlinear difference equation \[ y\left( z+1\right) =z^{\lambda /p}F\left( z^{1/p},y\left( z\right) \right) , \tag{1} \] where \(p\in \mathbb{N}\), \(\lambda \in \mathbb{Z}\) and \(F\) is a \(\mathbb{C} ^{n}\)-valued function, analytic in a neighbourhood of \(\left( \infty ,y_{0}\right) \) for some \(y_{0}\in \mathbb{C}^{n}\). The main result shows that the formal solution of (1) is accelero-summable (in some weaker form than given by \textit{B.~L.~J. Braaksma, B. F. Faber} and \textit{G. K. Immink} [Pac. J. Math. 195, No.~1, 35--65 (2000; Zbl 1008.39003)]), even \(0\) is a singular direction of level \(1\).
scientific article; zbMATH DE number 5899946