Rooted maps on orientable surfaces, Riccati's equation and continued fractions

Abstract

We present a new approach in the study of rooted maps without regard to genus. We prove the existence of a new type of equation for the generating series of these maps enumerated with respect to edges and vertices. This is Riccati's equation. It seems to be the first time that such a differential equation appears in the enumeration of rooted maps. Solving this equation leads to different closed forms of the studied generating series. The most interesting consequence is a development of this generating function in a very nice continued fraction leading to a new equation generalizing the well-known Dyck equation for rooted planar trees. In a second part, we also obtain a differential equation for the generating series of rooted trees regardless of the genus, with respect to edges. This also leads to a continued fraction for the generating series of rooted genus independent trees and to an unexpected relation between both previous generating series of trees and rooted maps.