Enumeration of weighted quadrant walks: criteria for algebraicity and D-finiteness

TL;DR

This paper introduces a unified elliptic function approach to classify when generating functions for weighted quarter-plane walks are algebraic or D-finite, simplifying and unifying previous case-by-case analyses.

Contribution

It provides a general method to determine algebraicity and D-finiteness of generating functions for weighted walks using elliptic functions, unifying prior diverse techniques.

Findings

Unified proof for algebraicity criteria

Unified proof for D-finiteness criteria

Applicable to various weighted walk models

Abstract

In the field of enumeration of weighted walks confined to the quarter plane, it is known that the generating functions behave very differently depending on the chosen step set; in practice, the techniques used in the literature depend on the complexity of the counting series. In this paper we introduce a unified approach based on the theory of elliptic functions, which allows us to have a common proof of the characterisation of the algebraicity and D-finiteness of the generating functions.