New existence theorems are constructed for asymptotic solutions of linear ordinary differential equations of arbitrary order in the neighborhood of an irregular singularity of rank 1 with distinct characteristic values.
Let (1) be an equation of the above type and be the set of characteristic values of (1). The author builds on the base of a set of canonical sectors and proves the following main theorem. there exists a unique solution of (1) such that
as , uniformly in any closed sector properly interior to . Furthermore, this asymptotic expansion can be differentiated times under the same circumstances, and the solutions are linearly independent.