Ramified Asymptotic Expansions for Some Linear Initial Value Problem With Mahler Transforms

We focus on a linear partial differential initial value problem in complex time t and complex space combined with time acting Mahler transforms. Bounded holomorphic solutions on bounded sectors edged at the origin in time and on horizontal strips in space are constructed by means of infinite sums of Fourier–Laplace transforms. Each term of these sums is shown to possess a formal Puiseux series as asymptotic expansions in time t involving finitely many ramifications. The generating series of all these formal Puiseux expansions defines a so‐called Hahn series with rational support, stemming from the fact that, in the general case, the number of ramifications attached to each term of the series increases with its index of summation. MSC2010 Classification : 35C10, 35C15, 35C20, 35R10