Geodesic
Counting walks in a quadrant: a unified approach via boundary value problems
Journal of the European Mathematical Society, Tome 14 (2012) no. 3, pp. 749-777.

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The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this yields for the first time an explicit expression of the counting generating function. Moreover, the nature of the integrand of the integral formulations is shown to be directly dependent on the finiteness of a naturally attached group of birational transformations as well as on the sign of the covariance of the walk.
DOI : 10.4171/jems/317
Classification : 05-XX, 30-XX, 00-XX
Keywords: Lattice walk, counting generating function, boundary value problem, conformal mapping, Weierstrass elliptic function, Riemann surface, uniformization 
@article{JEMS_2012_14_3_a4,
     author = {Kilian Raschel},
     title = {Counting walks in a quadrant: a unified approach via boundary value problems},
     journal = {Journal of the European Mathematical Society},
     pages = {749--777},
     publisher = {mathdoc},
     volume = {14},
     number = {3},
     year = {2012},
     doi = {10.4171/jems/317},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/317/}
}
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Kilian Raschel. Counting walks in a quadrant: a unified approach via boundary value problems. Journal of the European Mathematical Society, Tome 14 (2012) no. 3, pp. 749-777. doi : 10.4171/jems/317. http://geodesic.mathdoc.fr/articles/10.4171/jems/317/

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