  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 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 $z^n \phi_n(z),$ where $\phi_n(0) = 1.$ 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.
