Department of Mathematics and Statistics, San Diego State University
We consider the second order differential equation
with a small parameter where is even with
respect to . It is known that it has two
formal solutions , where
is a formal series in powers of
whose coeffiecients are functions of x.
Dunster, Lutz and Schäfke showed in 1993 that one resp. both of these solutions are summable in certain directions if satisfies certain conditions, in particular concerning its x-domain.
After reviewing these known results, we give necessary conditions for the summability of one or both of the above formal solutions in terms of . The method of proof involves a certain inverse problem, i.e. the construction of a differential equation of the above form exhibiting a prescribed Stokes phenomenon with respect to .
This research is joint work with Reinhard Schäfke from the Université Louis Pasteur in Strasbourg, France.