Counting Walks in the Quarter Plane
The central question addressed in this paper is the nature of the series Q(x, y ; t). When is it algebraic? When is it D-finite (or holonomic)? Can these properties be derived from the functional equation itself?
Our first result is a new proof of an old theorem due to Kreweras, according to which one of these walk models has, for mysterious reasons, an algebraic generating function. Then, we provide a new proof of a holonomy criterion recently proved by M. Petkovšek and the author. In both cases, we work directly from the functional equation.
KeywordsFunctional Equation Power Series Formal Power Series Positive Part Lattice Path
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