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

Thomas Dreyfus, Charlotte Hardouin, Julien Roques, Michael F. Singer

Published 2017-02-15Version 1

In the present paper, we introduce a new approach, relying on differential 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.