Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

Alin Bostan, Frédéric Chyzak, Mark van Hoeij, Manuel Kauers, Lucien Pech
2017 European journal of combinatorics (Print)  
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
more » ... 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
doi:10.1016/j.ejc.2016.10.010 fatcat:bivi2veq6nbnvdy6jdntxa6ugi