On $k$-summable normal forms of vector fields with one zero eigenvalue

In this paper, we study normal forms of analytic saddle-nodes in $\mathbb C^{n+1}$ with any Poincar\'e rank $k\in \mathbb N$. The approach and the results generalize those of Bonckaert and De Maesschalck from 2008 that considered $k=1$. In particular, we introduce a Banach convolutional algebra that is tailored to study differential equations in the Borel plane of order $k$. One of the subtleties that we take care of in this paper, is that nontrivial Jordan blocks are allowed in the linear part of the vector field. We anticipate that our approach can stimulate new research and be used to study different normal forms in future work.