Quarter-Plane Lattice Paths with Interacting Boundaries: The Kreweras and Reverse Kreweras Models

Lattice paths in the quarter plane have led to a large and varied set of results recent years. One major project has been classification step sets according properties corresponding generating functions, this involved variety techniques, some highly intricate specialised. The famous Kreweras reverse walk models are two particularly interesting models, as they among only four cases which algebraic functions. Here we investigate how change when boundary interactions introduced. That is, associate three real-valued weights $a,b,c$ with visits by walks $x$-axis, $y$-axis origin $(0,0)$ respectively. These were partially solved paper Beaton, Owczarek Rechnitzer (2019). We apply kernel method completely solve these models. find that an function for all $a,b,c$, regardless whether restricted end at or on one axes, may anywhere all. For walks, returning is algebraic, but other D-finite. To our knowledge first example quarter-plane model property.