Gevrey Order and Summability of Formal Series Solutions of some Classes of Inhomogeneous Linear Partial Differential Equations with Variable Coefficients

We investigate Gevrey order and summability properties of formal power series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients and analytic initial conditions. In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient k, in a given direction.