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Differential algebraic generating series of weighted walks in the quarter plane

Thomas Dreyfus

Abstract

In the present paper we study the nature of the trivariate generating series of weighted walks in the quarter plane. Combining the results of this paper to previous ones, we complete the proof of the following theorem. The series satisfies a nontrivial algebraic differential equation in one of its variable, if and only if it satisfies a nontrivial algebraic differential equation in each of its variables.

Differential algebraic generating series of weighted walks in the quarter plane

Abstract

In the present paper we study the nature of the trivariate generating series of weighted walks in the quarter plane. Combining the results of this paper to previous ones, we complete the proof of the following theorem. The series satisfies a nontrivial algebraic differential equation in one of its variable, if and only if it satisfies a nontrivial algebraic differential equation in each of its variables.

Paper Structure

This paper contains 12 sections, 24 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Kernel of the walk
  3. Functional equation
  4. Degenerate cases.
  5. Genus of the walk
  6. Group of the walk
  7. Uniformization of the curve
  8. Meromorphic continuation of the generating function
  9. Preliminary results on differential algebraicity
  10. Differential algebraicity of the generating function
  11. Finite group case
  12. Infinite group case

Key Result

Theorem 1.1

The following facts are equivalent:

Figures (2)

  • Figure 1: Our four nondegenerate models
  • Figure 2: The maps $i_{1}$ and $i_{2}$ restricted to the kernel curve $\overline{E}_{t}$ are denoted by $\iota_1$ and $\iota_2$, respectively.

Theorems & Definitions (49)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 2.2: Degenerate model
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • ...and 39 more