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

Ivo J. B. F. Adan, Johan S. H. van Leeuwaarden, Kilian Raschel

Published 2011-01-04, updated 2012-10-07Version 2

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.