An algorithm to obtain formal solutions of a linear homogeneous differential equation at an irregular singular point

Abstract

The algorithm DELIRE (Linear Differential Equation Irregular and REgular singularities) presented in this paper gives the formal solutions (normal and sub-normal) at an irregular singular point of a linear homogeneous differential operator, whose coefficients are formal Puiseux series [11]. This algorithm (greatly inspired by recent works of B. Malgrange) is based on the use of the Newton polygon of the differential operator studied. The part of DELIRE which gives the solutions in the neighborhood of singular regular points has been presented in [3]. DELIRE is included in the general algorithm said DESIR (Differential Equation System Irregular and Regular singularities), which deals with linear homogeneous differential system in the complex plan.