Algorithms for the splitting of formal series; applications to alien differential calculus

We present algorithms which involve both the splitting of formal series solutions to linear ordinary differential equations with polynomial coefficients into a finite sum of subseries which themselves will be solutions of linear ODEs, and the simplification of the recurrence relations satisfied by their coefficients.When coping with series that are solutions of a given differential equation at an irregular – singular point of rank k ≥ 2, it enables us to reduce the computations to series solutions of an ODE with an irregularity of rank one. In particular, we are able to conduct effective calculations with Écalle’s alien derivations for these series. We apply our techniques to some “accelerating functions” of Écalle.