M. Abramowitz and I. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, 1964.

A. Bostan and M. Kauers, The complete generating function for Gessel walks is algebraic, Proc. Amer, pp.3063-3078, 2010.
DOI : 10.1090/S0002-9939-2010-10398-2

URL : https://hal.archives-ouvertes.fr/hal-00780429

A. Bostan, I. Kurkova, and K. Raschel, A human proof of Gessel???s lattice path conjecture, Transactions of the American Mathematical Society, vol.369, issue.2, pp.1309-1023, 2013.
DOI : 10.1090/tran/6804

A. Bostan, K. Raschel, and B. Salvy, Non-D-finite excursions in the quarter plane, Journal of Combinatorial Theory, Series A, vol.121, pp.45-63, 2014.
DOI : 10.1016/j.jcta.2013.09.005

URL : https://hal.archives-ouvertes.fr/hal-00697386

M. Bousquet-mélou, Walks in the quarter plane: Kreweras??? algebraic model, The Annals of Applied Probability, vol.15, issue.2, pp.1451-1491, 2005.
DOI : 10.1214/105051605000000052

M. Bousquet-mélou and M. Mishna, Walks with small steps in the quarter plane, Contemp. Math, vol.520, pp.1-40, 2010.
DOI : 10.1090/conm/520/10252

M. Bousquet-mélou and M. Petkovsek, Walks confined in a quadrant are not always D-finite, Theoretical Computer Science, vol.307, issue.2, pp.257-276, 2003.
DOI : 10.1016/S0304-3975(03)00219-6

B. Chabat, Introduction à l'analyse complexe, 1990.

G. Fayolle, R. Iasnogorodski, and V. Malyshev, Random walks in the quarter-plane, 1999.
URL : https://hal.archives-ouvertes.fr/inria-00572276

G. Fayolle and K. Raschel, On the holonomy or algebraicity of generating functions counting lattice walks in the quarter-plane. Markov Process, pp.485-496, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00559676

G. Fayolle and K. Raschel, Random walks in the quarter plane with zero drift: an explicit criterion for the finiteness of the associated group. Markov Process, pp.619-636, 2011.
URL : https://hal.archives-ouvertes.fr/inria-00572276

P. Flajolet and R. Sedgewick, Analytic combinatorics, 2009.
DOI : 10.1017/CBO9780511801655

URL : https://hal.archives-ouvertes.fr/inria-00072739

L. Flatto and S. Hahn, Two Parallel Queues Created by Arrivals with Two Demands I, SIAM Journal on Applied Mathematics, vol.44, issue.5, pp.1041-1053, 1984.
DOI : 10.1137/0144074

L. Flatto, Two Parallel Queues Created by Arrivals with Two Demands I, SIAM Journal on Applied Mathematics, vol.44, issue.5, pp.861-878, 1985.
DOI : 10.1137/0144074

G. Jones and D. Singerman, Complex Functions, 1987.
DOI : 10.1017/CBO9781139171915

M. Kauers, C. Koutschan, and D. Zeilberger, Proof of Ira Gessel's lattice path conjecture, Proceedings of the National Academy of Sciences, vol.106, issue.28, pp.11502-11505, 2009.
DOI : 10.1073/pnas.0901678106

G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du B.U.R.O, vol.6, pp.5-105, 1965.

I. Kurkova and K. Raschel, Explicit expression for the generating function counting Gessel??s walks, Advances in Applied Mathematics, vol.47, issue.3, pp.414-433, 2011.
DOI : 10.1016/j.aam.2010.11.004

I. Kurkova and K. Raschel, On the functions counting walks with small steps in the quarter plane, Publications math??matiques de l'IH??S, vol.14, issue.1, pp.69-114, 2012.
DOI : 10.1007/s10240-012-0045-7

URL : https://hal.archives-ouvertes.fr/hal-00628424

V. Malyshev, Random Walks, Wiener-Hopf Equations in the Quarter Plane, Galois Automorphisms (in Russian), 1970.

V. Malyshev, Positive random walks and Galois theory, Uspehi Matem. Nauk, vol.26, pp.227-228, 1971.

V. Malyshev, An analytical method in the theory of two-dimensional positive random walks. Siberian Math, J, vol.13, pp.1314-1329, 1972.

S. Melczer and M. Mishna, Singularity Analysis Via the Iterated Kernel Method, Combinatorics, Probability and Computing, vol.17, issue.05, 2013.
DOI : 10.1016/S0304-3975(02)00007-5

URL : https://hal.archives-ouvertes.fr/hal-01394160

M. Mishna and A. Rechnitzer, Two non-holonomic lattice walks in the quarter plane, Theoretical Computer Science, vol.410, issue.38-40, pp.3616-3630, 2009.
DOI : 10.1016/j.tcs.2009.04.008

K. Raschel, Counting walks in a quadrant: a unified approach via boundary value problems, Journal of the European Mathematical Society, vol.14, pp.749-777, 2012.
DOI : 10.4171/JEMS/317

URL : https://hal.archives-ouvertes.fr/hal-00461853

G. Sansone and J. And-gerretsen, Lectures on the theory of functions of a complex variable. I. Holomorphic functions, P. Noordhoff, 1960.

G. Watson, W. , and E. , A course of modern analysis, 1962.

P. Jussieu, 75252 Paris Cedex 05, France E-mail address: Irina.Kourkova@upmc.fr CNRS & Fédération de recherche Denis Poisson & Laboratoire de Mathématiques et Physique Théorique