ON THE NATURE OF THE GENERATING SERIES OF WALKS IN THE QUARTER PLANE

Thomas Dreyfus (1) , Charlotte Hardouin (2) , Julien Roques (3) , Michael F. Singer (4)

1 IRMA - Institut de Recherche Mathématique Avancée
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 IF - Institut Fourier
4 NCSU - Department of Mathematics [Raleigh]

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 coefficients.

Mots clés: Transcendence; Difference Galois theory; Random walks; Elliptic functions

Domaines: Mathématiques générales [math.GM]

Thomas Dreyfus, Charlotte Hardouin, Julien Roques, Michael F. Singer. ON THE NATURE OF THE GENERATING SERIES OF WALKS IN THE QUARTER PLANE. Inventiones Mathematicae, 2018, 213 (1), pp.139 - 203. ⟨10.1007/s00222-018-0787-z⟩. ⟨hal-01897314⟩