A divertent generating function that can be summed and analysed analytically

We study a recurrence relation, where the generating function, as a formal power series, satisfies a differential equation that can be solved in a suitable domain; this yields an analytic function in a domain, but the solution is singular at the origin and the generating function has radius of convergence 0. Nevertheless, the solution to the recurrence can be obtained from the analytic solution by finding an asymptotic series expansion. Conversely, the analytic solution can be obtained by summing the generating function by the Borel summation method. Note I have been informed by several participants at the colloquium for Philippe Flajolet’s 60th birthday that many or all results of this paper are known. This is thus a draft that requires revision, including addition of several more references, before a possible publication.