On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations

We consider a class of $n^{\text{th}}$-order linear ordinary differential equations with a large parameter u. Analytic solutions of these equations can be described by (divergent) formal series in descending powers of u. We demonstrate that, given mild conditions on the potential functions of the equation, the formal solutions are Borel summable with respect to the parameter u in large, unbounded domains of the independent variable. We establish that the formal series expansions serve as asymptotic expansions, uniform with respect to the independent variable, for the Borel re-summed exact solutions. Additionally, we show that the exact solutions can be expressed using factorial series in the parameter, and these expansions converge in half-planes, uniformly with respect to the independent variable. To illustrate our theory, we apply it to an $n^{\text{th}}$-order Airy-type equation.