The compensation approach for walks with small steps in the quarter plane

TL;DR

This paper introduces the first application of the compensation approach to counting lattice walks with small steps in the quarter plane, providing explicit generating functions and asymptotic analysis.

Contribution

It applies the compensation approach to a new class of walks in the quarter plane, deriving explicit meromorphic generating functions.

Findings

Generated explicit counting functions for the walks.

Proved the generating functions are meromorphic and nonholonomic.

Provided asymptotic expressions for counting coefficients.

Abstract

This paper is the first application of the compensation approach to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane $Z_{+}^{2}$ with a step set that is a subset of $\{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}$ in the interior of $Z_{+}^{2}$. We derive an explicit expression for the counting generating function, which turns out to be meromorphic and nonholonomic, can be easily inverted, and can be used to obtain asymptotic expressions for the counting coefficients.