B. Adamczewski and C. Faverjon, M??thode de Mahler : relations lin??aires, transcendance et applications aux nombres automatiques, Proceedings of the London Mathematical Society, vol.42, issue.1, pp.55-90, 2017.
DOI : 10.1112/plms/s2-42.1.230

URL : http://arxiv.org/abs/1508.07158

B. Adamczewski and C. Faverjon, Méthode de Mahler, transcendance et relations linéaires : aspects effectifs, J. Théor. Nombres Bordeaux
DOI : 10.1112/plms.12038

URL : http://arxiv.org/abs/1508.07158

Y. André, Series Gevrey de Type Arithmetique, I. Theoremes de Purete et de Dualite, The Annals of Mathematics, vol.151, issue.2, pp.705-740, 2000.
DOI : 10.2307/121045

Y. André, Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence, Annales scientifiques de l'??cole normale sup??rieure, vol.47, issue.2, pp.449-467, 2014.
DOI : 10.24033/asens.2218

M. Barkatou, On Rational Solutions of Systems of Linear Differential Equations, Journal of Symbolic Computation, vol.28, issue.4-5, pp.547-567, 1999.
DOI : 10.1006/jsco.1999.0314

V. V. Batyrev and D. Van-straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Communications in Mathematical Physics, vol.1, issue.2, pp.493-533, 1995.
DOI : 10.4310/SDG.1990.v1.n1.a5

URL : http://arxiv.org/abs/alg-geom/9307010

D. Bertrand and F. Beukers, ??quations diff??rentielles lin??aires et majorations de multiplicit??s, Annales scientifiques de l'??cole normale sup??rieure, vol.18, issue.1, pp.181-192, 1985.
DOI : 10.24033/asens.1488

D. Bertrand, V. Chirskii, and J. Yebbou, Effective estimates for global relations on Euler-type series, Annales de la Faculté des sciences de Toulouse : Mathématiques, Sér. 6, pp.241-260, 2004.
DOI : 10.5802/afst.1069

URL : http://www.numdam.org/article/AFST_2004_6_13_2_241_0.pdf

F. Beukers, A refined version of the Siegel???Shidlovskii theorem, Annals of Mathematics, vol.163, issue.1, pp.369-379, 2006.
DOI : 10.4007/annals.2006.163.369

S. Fischler and T. , On the values of $G$-functions, Commentarii Mathematici Helvetici, vol.89, issue.2, pp.313-341, 2014.
DOI : 10.4171/CMH/321

URL : https://hal.archives-ouvertes.fr/hal-00581002

S. Fischler and T. , Th??orie arithm??tique des E-op??rateurs, Journal de l?????cole polytechnique ??? Math??matiques, vol.3, pp.31-65, 2016.
DOI : 10.5802/jep.28

S. Fischler and T. , On the linear independence of values of G-functions, preprint, p.45, 2017.

D. Yu and . Grigoriev, Complexity of factoring and calculating the GCD of linear ordinary differential operators, J. Symb. Comput, vol.10, pp.7-37, 1990.

D. Yu, Grigoriev, private communication, 7th of, 2017.

M. Van-hoeij, Factorization of Differential Operators with Rational Functions Coefficients, Journal of Symbolic Computation, vol.24, issue.5, pp.537-561, 1997.
DOI : 10.1006/jsco.1997.0151

N. Katz, Exponential sums and differential equations, Annals of Mathematical Studies, vol.124, 1990.
DOI : 10.1515/9781400882434

W. Koepf, Hypergeometric summation: An algorithmic approach to summation and special function identities, 2014.

Y. V. Nesterenko and A. B. Shidlovskii, Linear independence of values of $ E$-functions, translated in english in Sb. Math, pp.93-108, 1996.
DOI : 10.1070/SM1996v187n08ABEH000152

T. Rivoal and J. Roques, Siegel's problem for E-functions of order 2, preprint, p.14, 2017.

P. Philippon, Groupes de Galois et nombres automatiques, Journal of the London Mathematical Society, vol.20, issue.28, pp.596-614, 2015.
DOI : 10.1007/978-3-662-29244-0

URL : https://hal.archives-ouvertes.fr/hal-01230858

K. Wegschaider, Computer Generated Proofs of Binomial Multi-Sum Identities, 1997.

B. Adamczewski, U. Lyon, and C. Umr, Institut Camille Jordan, F-69622 Villeurbanne Cedex, France. ? E-mail : Boris.Adamczewski@math.cnrs.fr Tanguy Rivoal, CS, vol.5208, issue.40700