*(English)*Zbl 0896.34049

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) $L\left(y\right)=0$ be an equation of the above type and ${\Lambda}$ be the set of characteristic values of (1). The author builds on the base of ${\Lambda}$ a set of canonical sectors $S=\{{S}_{\lambda}:\lambda \in {\Lambda}\}$ and proves the following main theorem. $\forall \lambda \in {\Lambda}$ there exists a unique solution ${w}_{\lambda}$ of (1) such that

as $z\to \infty $, uniformly in any closed sector properly interior to ${S}_{\lambda}$. Furthermore, this asymptotic expansion can be differentiated $n-1$ times under the same circumstances, and the $n$ solutions ${w}_{\lambda}$ are linearly independent.