Resurgent deformations for an ordinary differential equation of order 2 (Q868740)

The paper is the first of three papers to come. The authors consider the second order differential equation \[ (d^2/dx^2)\Phi (x)=(P_m(x)/x^2)\Phi (x) \] in the complex domain, where the monic polynomial \(P_m\) is of degree \(m\). They investigate the asymptotic and resurgent properties of its solutions at infinity, in particular -- the dependence of the Stokes-Sibuya multipliers (SSM) on the coefficients of \(P_m\). They derive a set of functional equations for the SSM (taking into account the nontrivial monodromy at the origin) and show how these equations can be used to compute the SSM for a class of polynomials \(P_m\). In particular, they obtain conditions for isomonodromic deformations when \(m=3\).