$x(1-t(x+x^{-1}))F(x;t) = x-tF(0;t)$

Abstract

The purpose of these notes is to introduce some of the problems the enumeration of lattice walks is dedicated to and familiarize with some of the arguments they can be addressed with. We discuss the enumeration of lattice walks, their generating functions, and the functional equations they satisfy. We focus on algebraic methods for manipulating and solving these equations. Elementary power series algebra plays a prominent role, computer algebra too, but we repeatedly digress and present ideas and methods of different kind whenever it is appropriate. The exposition is organized around the most simple yet non-trivial problem: the enumeration of simple walks on the half-line. The intention is to illustrate different techniques without getting technical.