On some hyperelliptic Hurwitz-Hodge integrals

This short note addresses Hodge integrals over the hyperelliptic locus. Recently Afandi computed, via localisation techniques, such one-descendant integrals and showed that they are Stirling numbers. We give another proof of the same statement by a very short argument, exploiting Chern classes of spin structures and relations arising from Topological Recursion in the sense of Eynard and Orantin. These techniques seem also suitable to deal with three orthogonal generalisations: 1. the extension to the r-hyperelliptic locus, 2. the extension to an arbitrary number of non-Weierstrass pairs of points, 3. the extension to multiple descendants.