A functional equation with Borel summable solutions and irregular singular solutions

Abstract

A Functional equation ∑_𝑖=1^𝑚𝑎_𝑖(𝑧)𝑢(𝜑_𝑖(𝑧))=𝑓(𝑧) is considered. First we show the existence of solutions of formal power series. Second we study the homogeneous equation (𝑓(𝑧)≡ 0) and construct formal solutions containing exponential factors. Finally it is shown that there exists a genuine solution in a sector whose asymptotic expansion is a formal solution, by using the theory of Borel summability of formal power series. The equation has similar properties to those of irregular singular type in the theory of ordinary differential equations.