Werner Balser

From Divergent Power Series to Analytic Functions

Asymptotic power series.- Laplace and borel transforms.- Summable power series.- Cauchy-Heine transform.- Acceleration operators.- Multisummable power series.- Some equivalent definitions of multisummability.- Formal solutions to non-linear ODE.

On the Convergence of Borel Approximants

For a wide class of functions having a representation as Laplace integrals, we obtain expansions in terms of certain higher transcendental functions. Compared with the classical method of expansions as factorial series, these series converge considerably faster and, at the same time, in larger regions.

Existence and structure of complete formal solutions of non-linear meromorphic systems of ordinary differential equations

For general non-linear n-dimensional homogeneous systems of meromorphic ODE, we show existence of a complete formal solution, formally depending upon n free parameters. We investigate the structure of such formal solutions and, in particular, obtain a factorization, analogous to results of J.-P. Ramis and (independently) the author for linear systems.

Computation of formal fundamental solutions

Abstract A new proof for the existence of formal fundamental solutions of meromorphic systems of linear ordinary differential equations is given. The proof is presented in a totally constructive form. Moreover, we give a corresponding proof of existence for highest level formal solutions.

Moment methods and formal power series

Abstract We study moment methods of summation and investigate their properties, especially with regard to summation of (formal) power series. In particular, we obtain a generalization of Ecalles multisummability method, which has attracted much attention lately.

Multisummability of complete formal solutions for non-linear systems of Meromorphic ordinary differential equations

In an earlier paper of the authors, existence of a complete formal solution for non-linear systems has been shown. Here, we investigate their multi-summability properties.

First-level formal solutions and multisummability

The concept of first-level formal solutions has been introduced and studied in earlier papers of the author. Here, a different but equivalent definition is given, and their existence proof is outlined. Furthermore, it is shown how this concept leads to a proof for the multisummability of formal solutions (in the usual sense) for linear systems of ODE.