@tpfto@mathstodon.xyz Looks like you can generalize Burmann's theorem by generalizing the series inversion theorem. This article includes the case of phi having a fixed number of zero derivatives at a. https://www.mathematica-journal.com/2014/11/24/on-burmanns-theorem-and-its-application-to-problems-of-linear-and-nonlinear-heat-transfer-and-diffusion/