Inventiones mathematicae

, Volume 213, Issue 1, pp 139–203 | Cite as

On the nature of the generating series of walks in the quarter plane

  • Thomas Dreyfus
  • Charlotte Hardouin
  • Julien RoquesEmail author
  • Michael F. Singer


In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many of the recent results about these series, but also to go beyond them. For instance, we give for the first time hypertranscendency results, i.e., we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational function coefficients.

Mathematics Subject Classification

05A15 30D05 39A06 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Thomas Dreyfus
    • 1
  • Charlotte Hardouin
    • 2
  • Julien Roques
    • 3
    Email author
  • Michael F. Singer
    • 4
  1. 1.Institut de Recherche Mathématique Avancée, U.M.R. 7501Université de Strasbourg et C.N.R.S.StrasbourgFrance
  2. 2.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouseFrance
  3. 3.Institut Fourier, CNRS UMR 5582Université Grenoble AlpesSt Martin d’HèresFrance
  4. 4.Department of MathematicsNorth Carolina State UniversityRaleighUSA

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