Hypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter Plane

TL;DR

This paper proves that the generating functions for 19 specific quarter-plane lattice walk models are solutions to differential equations and can be expressed via hypergeometric functions, revealing their transcendental nature and algebraic specializations.

Contribution

It provides the first rigorous proof that these 19 models' generating functions satisfy specific differential equations and are expressible through hypergeometric functions.

Findings

All 19 generating functions satisfy linear differential equations.

They can be expressed in terms of Gauss' hypergeometric functions.

Most of these functions are transcendental, with only four algebraic cases.

Abstract

We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on $\mathbb{Z}^2$ defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or $-1$. We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane $\mathbb{N}^2$, counted by their length and by the position of their ending point. Bousquet-M\'elou and Mishna [Contemp. Math., pp. 1--39, Amer. Math. Soc., 2010] identified 19 models of walks that possess a D-finite generating function; linear differential equations have then been guessed in these cases by Bostan and Kauers [FPSAC 2009, Discrete Math. Theor. Comput. Sci. Proc., pp. 201--215, 2009]. We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their $19 \times 4$ combinatorially meaningful specializations only four are algebraic functions.