Lattice and Schrِder paths with periodic boundaries

We consider paths in the plane with ( 1 , 0 ), ( 0 , 1 ), and ( a , b )-steps that start at the origin, end at height n , and stay strictly to the left of a given non-decreasing right boundary. We show that if the boundary is periodic and has slope at most b / a , then the ordinary generating function for the number of such paths ending at height n is algebraic. Our argument is in two parts. We use a simple combinatorial decomposition to obtain an Appell relation or “umbral” generating function, in which the power z n is replaced by a power series of the form z n φ n ( z ) , where φ n ( 0 ) = 1 . Then we convert (in an explicit way) the umbral generating function to an ordinary generating function by solving a system of linear equations and a polynomial equation. This conversion implies that the ordinary generating function is algebraic. We give several concrete examples, including an alternative way to solve the tennis ball problem.