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

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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).

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Authors and Affiliations

  1. Laboratoire de Probabilités et, Modèles Aléatoires, Université Pierre et Marie Curie, 4 Place Jussieu, 75252, Paris Cedex 05, France

    I. Kurkova

  2. CNRS & Fédération de recherche Denis Poisson & Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37200, Tours, France

    K. Raschel

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  1. I. Kurkova
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  2. K. Raschel
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Correspondence to I. Kurkova.

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Kurkova, I., Raschel, K. New Steps in Walks with Small Steps in the Quarter Plane: Series Expressions for the Generating Functions. Ann. Comb. 19, 461–511 (2015). https://doi.org/10.1007/s00026-015-0279-4

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  • DOI: https://doi.org/10.1007/s00026-015-0279-4

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