Quadrant Walks Starting Outside the Quadrant

TL;DR

This paper explores a functional equation related to lattice walks in the plane, revealing solutions that are transcendental or possibly non-D-finite, and introduces a novel walk model with restricted axis crossings.

Contribution

It introduces a new lattice walk model with unique boundary restrictions and analyzes its generating function, showing it is transcendental or non-D-finite, unlike typical algebraic solutions.

Findings

The generating function is not algebraic despite zero orbit sum.

Solutions for variants are transcendental or possibly non-D-finite.

The model involves walks starting at (-1,-1) with restricted axis crossings.

Abstract

We investigate a functional equation which resembles the functional equation for the generating function of a lattice walk model for the quarter plane. The interesting feature of this equation is that its orbit sum is zero while its solution is not algebraic. The solution can be interpreted as the generating function of lattice walks in $\mathbb{Z}^2$ starting at $(-1,-1)$ and subject to the restriction that the coordinate axes can be crossed only in one direction. We also consider certain variants of the equation, all of which seem to have transcendental solutions. In one case, the solution is perhaps not even D-finite.