Appears in collection : Algebra, arithmetic and combinatorics of differential and difference equations / Algèbre, arithmétique et combinatoire des équations différentielles et aux différences

Excursions are walks which start and end at prescribed locations. In this talk we consider the counting sequences of excursions, more precisely, the functional equations their generating functions satisfy. We focus on two sources of excursion problems: walks defined by their allowable steps, taken on integer lattices restricted to cones; and walks on Cayley graphs with a given set of generators. The latter is related to the cogrowth problems of groups. In both cases we are interested in relating the nature of the generating function (i.e. rational, algebraic, D-finite, etc.) and combinatorial properties of the models. We are also interested in the relation between the excursions, and less restricted families of walks. Please note: A few corrections were made to the PDF file of this talk, the new version is available at the bottom of the page.