S. A. Abramov, K. Yu, and . Kvashenko, Fast algorithms to search for the rational solutions of linear differential equations with polynomial coefficients, Proceedings of the 1991 international symposium on Symbolic and algebraic computation , ISSAC '91, pp.267-270, 1991.
DOI : 10.1145/120694.120735

S. A. Abramov and M. Van-hoeij, Integration of solutions of linear functional equations, Integral Transforms and Special Functions, vol.34, issue.6, pp.3-12, 1999.
DOI : 10.1080/10652469908819212

G. Almkvist and D. Zeilberger, The method of differentiating under the integral sign, Journal of Symbolic Computation, vol.10, issue.6, pp.571-591, 1990.
DOI : 10.1016/S0747-7171(08)80159-9

A. Moulay and . Barkatou, On rational solutions of systems of linear differential equations, Differential algebra and differential equations, pp.4-5547, 1999.

I. N. Bern?te?-in, Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients, Funct. Anal. Appl.Russian, vol.5, issue.2, pp.1-16, 1971.

I. N. Bern?te?-in, The analytic continuation of generalized functions with respect to a parameter, Funct. Anal. Appl.Russian, vol.6, issue.4, pp.26-40, 1972.

F. Beukers, Irrationality of ? 2 , periods of an elliptic curve and ? 1 (5), Diophantine approximations and transcendental numbers (Luminy, pp.47-66, 1982.

A. Bostan, F. Chyzak, and B. Salvy, Grégoire Lecerf, and Éric Schost. Differential equations for algebraic functions, pp.25-32, 2007.

A. Bostan and M. Kauers, Automatic classification of restricted lattice walks, DMTCS Proceedings of the 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC'09), pp.203-217, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00780428

F. Chyzak, An extension of Zeilberger's fast algorithm to general holonomic functions, Discrete Mathematics, vol.217, issue.1-3, pp.115-134, 1997.
DOI : 10.1016/S0012-365X(99)00259-9

F. Chyzak, M. Kauers, and B. Salvy, A non-holonomic systems approach to special function identities, Proceedings of the 2009 international symposium on Symbolic and algebraic computation, ISSAC '09, pp.111-118, 2009.
DOI : 10.1145/1576702.1576720
URL : https://hal.archives-ouvertes.fr/inria-00376526

R. Debeerst, Solving differential equations in terms of bessel functions, Proceedings of the twenty-first international symposium on Symbolic and algebraic computation, ISSAC '08, 2007.
DOI : 10.1145/1390768.1390777

R. Debeerst, M. Van-hoeij, and W. Koepf, Solving differential equations in terms of bessel functions, Proceedings of the twenty-first international symposium on Symbolic and algebraic computation, ISSAC '08, pp.39-46, 2008.
DOI : 10.1145/1390768.1390777

B. Dwork, Differential Operators with Nilpotent p-Curvature, American Journal of Mathematics, vol.112, issue.5, pp.749-786, 1990.
DOI : 10.2307/2374806

M. Erickson, S. Fernando, and K. Tran, Enumerating Rook and Queen Paths, Bull. Inst. Combin. Appl, vol.60, pp.37-48, 2010.

M. C. Fasenmyer, Some generalized hypergeometric polynomials, Bulletin of the American Mathematical Society, vol.53, issue.8, pp.806-812, 1947.
DOI : 10.1090/S0002-9904-1947-08893-5

P. Flajolet, Analytic models and ambiguity of context-free languages, Twelfth international colloquium on automata, languages and programming, pp.283-309, 1985.
DOI : 10.1016/0304-3975(87)90011-9
URL : https://hal.archives-ouvertes.fr/inria-00076071

P. Flajolet and A. Odlyzko, Singularity Analysis of Generating Functions, SIAM Journal on Discrete Mathematics, vol.3, issue.2, pp.216-240, 1990.
DOI : 10.1137/0403019
URL : https://hal.archives-ouvertes.fr/inria-00075725

P. Flajolet and R. Sedgewick, Analytic combinatorics, 2009.
DOI : 10.1017/CBO9780511801655
URL : https://hal.archives-ouvertes.fr/inria-00072739

É. Goursat, Sur l'??quation diff??rentielle lin??aire, qui admet pour int??grale la s??rie hyperg??om??trique, Annales scientifiques de l'??cole normale sup??rieure, vol.10, issue.2, pp.3-142
DOI : 10.24033/asens.207

E. L. Ince, Ordinary Differential Equations, The Mathematical Gazette, vol.18, issue.231, 1956.
DOI : 10.2307/3605524

C. Koutschan, A Fast Approach to Creative Telescoping, Mathematics in Computer Science, vol.32, issue.3, pp.259-266, 2010.
DOI : 10.1007/s11786-010-0055-0

L. Lipshitz, The diagonal of a D-finite power series is D-finite, Journal of Algebra, vol.113, issue.2, pp.373-378, 1988.
DOI : 10.1016/0021-8693(88)90166-4

L. Lipshitz, D-finite power series, Journal of Algebra, vol.122, issue.2, pp.353-373, 1989.
DOI : 10.1016/0021-8693(89)90222-6

M. Petkov?ek, Hypergeometric solutions of linear recurrences with polynomial coefficients, Journal of Symbolic Computation, vol.14, issue.2-3, pp.243-264, 1992.
DOI : 10.1016/0747-7171(92)90038-6

A. Raichev and M. C. Wilson, A new method for computing asymptotics of diagonal coefficients of multivariate generating functions, Conference on Analysis of Algorithms, 2007.
URL : https://hal.archives-ouvertes.fr/hal-01184779

H. Schwarz, Ueber diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, J. Reine Angew. Math, vol.75, pp.292-335, 1873.

F. Michael and . Singer, Solving homogeneous linear differential equations in terms of second order linear differential equations, Amer. J. Math, vol.107, issue.3, pp.663-696, 1985.

R. P. Stanley, Differentiably Finite Power Series, European Journal of Combinatorics, vol.1, issue.2, pp.175-188, 1980.
DOI : 10.1016/S0195-6698(80)80051-5
URL : http://doi.org/10.1016/s0195-6698(80)80051-5

R. P. Stanley, Enumerative combinatorics, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin, 1999.

N. Takayama, An approach to the zero recognition problem by Buchberger algorithm, Journal of Symbolic Computation, vol.14, issue.2-3, pp.265-282, 1992.
DOI : 10.1016/0747-7171(92)90039-7

K. Takeuchi, Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo, vol.24, pp.201-212, 1977.

M. Van-hoeij, Implementation, 2001.

M. Van-hoeij, Solving third order linear differential equations in terms of second order equations, Proceedings of the 2007 international symposium on Symbolic and algebraic computation , ISSAC '07, pp.355-360, 2007.
DOI : 10.1145/1277548.1277596

M. Van-hoeij and Q. Yuan, Finding all bessel type solutions for linear differential equations with rational function coefficients, Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ISSAC '10, pp.37-44, 2010.
DOI : 10.1145/1837934.1837948

R. Vid¯-unas, Algebraic Transformations of Gauss Hypergeometric Functions, Funkcialaj Ekvacioj, vol.52, issue.2, pp.139-180, 2009.
DOI : 10.1619/fesi.52.139

W. Wasow, Asymptotic expansions for ordinary differential equations, 1987.

D. Zeilberger, Sister Celine's technique and its generalizations, Journal of Mathematical Analysis and Applications, vol.85, issue.1, pp.114-145, 1982.
DOI : 10.1016/0022-247X(82)90029-4
URL : http://doi.org/10.1016/0022-247x(82)90029-4

D. Zeilberger, A fast algorithm for proving terminating hypergeometric identities, Discrete Mathematics, vol.80, issue.2, pp.207-211, 1990.
DOI : 10.1016/0012-365X(90)90120-7

D. Zeilberger, A holonomic systems approach to special functions identities, Journal of Computational and Applied Mathematics, vol.32, issue.3, pp.321-368, 1990.
DOI : 10.1016/0377-0427(90)90042-X

D. Zeilberger, The method of creative telescoping, Journal of Symbolic Computation, vol.11, issue.3, pp.195-204, 1991.
DOI : 10.1016/S0747-7171(08)80044-2