Resurgent Deformations for an Ordinary Differential Equation of Order 2

We consider in the complex field the differential equation $\displaystyle \frac{d^2}{d x^2} \Phi(x) = \frac{P_m(x,\a)}{x^2}\Phi(x)$,where$P_m$ is a monic polynomial function of order $m$ with coefficients $\text{a}=(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_m)$. We investigate the asymptotic, resurgent, properties of the solutions at infinity, focusing in particular on the analytic dependence on $\text{a}$ of the Stokes-Sibuya multipliers. Taking into account the non trivial monodromy at the origin, we derive a set of functional equations for the Stokes-Sibuya multipliers. We show how these functional relations can be used to compute the Stokes multipliers for a class of polynomials $P_m$. In particular, we obtain conditions for isomonodromic deformations when $m=3$.