We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on Z 2 defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or −1. We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane N 2 , counted by their length and by the position of their ending point. Bousquet-Mélou and Mishna [Contemp. Math., pp. 1-39, Amer. Math. Soc., 2010] identified 19 models

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... f walks that possess a D-finite generating function; linear differential equations have then been guessed in these cases by Bostan and Kauers [FPSAC 2009 , Discrete Math. Theor. Comput. Sci. Proc., pp. 201-215, 2009]. We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their 19 × 4 combinatorially meaningful specializations only four are algebraic functions. The generating function Q(t) corresponding to the example step set S above is algebraic [10, 22, 31], i.e., it satisfies a polynomial equation P (t, Q(t)) = 0 for some P ∈ Q[t, T ] \ {0}. But this is not the case for all other step sets. Still, among those step sets that induce a transcendental (i.e., non-algebraic) generating function Q(t), some have a Q(t) that is D-finite, i.e., that satisfies a linear differential equation with polynomial coefficients. The step set S = {(1, 1), (−1, 1), (0, −1)} is an example for this case [9, 12] . Finally, there are also step sets whose corresponding generating function is not even D-finite; Mishna and Rechnitzer [38] proved that this is the case for example when S