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Annals of Combinatorics

, Volume 19, Issue 3, pp 461–511 | Cite as

New Steps in Walks with Small Steps in the Quarter Plane: Series Expressions for the Generating Functions

  • I. KurkovaEmail author
  • K. Raschel
Article

Abstract

In this article we obtain new expressions for the generating functions counting (nonsingular) walks with small steps in the quarter plane. Those are given in terms of infinite series, while in the literature, the standard expressions use solutions to boundary value problems. We illustrate our results with three examples (an algebraic case, a transcendental D-finite case, and an infinite group model).

Keywords

walks in the quarter plane counting generating function holonomy group of the walk Riemann surface elliptic functions uniformization universal covering 

Mathematics Subject Classification

05A15 30F10 30D05 

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© Springer Basel 2015

Authors and Affiliations

  1. 1.Laboratoire de Probabilités et, Modèles AléatoiresUniversité Pierre et Marie CurieParis Cedex 05France
  2. 2.CNRS & Fédération de recherche Denis Poisson & Laboratoire de Mathématiques et Physique ThéoriqueUniversité de ToursToursFrance

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