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The aim of this article is to introduce a unified method to obtain explicitintegral representations of the trivariate generating function counting thewalks with small steps which are confined to a quarter plane. For many models,this yields for the first time an explicit expression of the countinggenerating function. Moreover, the nature of the integrand of the integralformulations is shown to be directly dependent on the finiteness of a naturallyattached group of birational transformations as well as on the sign of thecovariance of the walk

Counting walks in a quadrant: a unified approach via boundary value problems