Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks

TL;DR

This paper investigates the algebraic and differential transcendence properties of generating functions for weighted quadrant walks, using advanced techniques like differential Galois theory and complex analysis to classify their nature.

Contribution

It provides new criteria for algebraicity and transcendence of generating functions in weighted quadrant walks, extending previous results to the weighted case with novel methods.

Findings

Certain generating functions are algebraic depending on step probabilities.

Some generating functions do not satisfy any algebraic differential equation.

Extended key intermediate results to the weighted case, including analytic continuation.

Abstract

We consider weighted small step walks in the positive quadrant, and provide algebraicity and differential transcendence results for the underlying generating functions: we prove that depending on the probabilities of allowed steps, certain of the generating functions are algebraic over the field of rational functions, while some others do not satisfy any algebraic differential equation with rational function coefficients. Our techniques involve differential Galois theory for difference equations as well as complex analysis (Weierstrass parameterization of elliptic curves). We also extend to the weighted case many key intermediate results, as a theorem of analytic continuation of the generating functions.