, Precomputation, independent of n 0

, decreasing order: a. Compute f

, > sup fF [ (n)] (p / q; k) : k > n + 1; k 2 / Z g (cf. (37)) using Algorithm 22, and set b = max (b 1 Otherwise

, If b is larger than the maximum of the S(k) dened so far, set S(n) = b

, Find the smallest n > n 0 on which S is dened

, There is no harm in also replacing () by zero in (30) when we take (n) = 1 because the origin is an ordinary point: doing so only makes the rst few terms of the bound innite

R. Barrio, M. Rodríguez, A. Abad, and F. Blesa, Breaking the limits: The Taylor series method, Applied Mathematics and Computation, vol.217, issue.20, 2011.
DOI : 10.1016/j.amc.2011.02.080

A. Bostan and É. Schost, Polynomial evaluation and interpolation on special sets of points, Journal of Complexity, vol.21, issue.4, p.420446, 2005.
DOI : 10.1016/j.jco.2004.09.009

P. Richard, P. Brent, and . Zimmermann, Modern Computer Arithmetic, 2010.

D. Broadhurst, Bessel moments, random walks and Calabi-Yau equations, 2009.

A. Cauchy, Mémoire sur l'emploi du nouveau calcul, appelé calcul des limites, dans l'intégration d'un système d'équations diérentielles. Comptes-rendus de l'Académie des Sciences, 15:14, jul 1842, pp.5-17

A. L. and C. , Oeuvres complètes d'Augustin Cauchy, Ière série, tome VII, 1892.

V. David, G. V. Chudnovsky, and . Chudnovsky, Computer algebra in the service of mathematical physics and number theory, p.109232

V. David, R. D. Chudnovsky, and . Jenks, Computers in Mathematics Talks from the International Conference on Computers and Mathematics, 1986.

J. H. Davenport and M. Mignotte, On nding the largest root of a polynomial, Modélisation Mathématique et Analyse Numérique, vol.24, p.693696, 1990.

, Digital library of mathematical functions Companion to the NIST Handbook of Mathematical Functions, 2010.

Z. Du and C. Yap, Uniform complexity of approximating hypergeometric functions with absolute error, Proceedings of the 7th Asian Symposium on Computer Mathematics (ASCM 2005), pages 246249. Korea Institute for Advanced Study, 2005.

I. Eble and M. Neher, ACETAF, ACM Transactions on Mathematical Software, vol.29, issue.3, p.263286, 2003.
DOI : 10.1145/838250.838252

F. G. and F. , Über die Integration der linearen Dierentialgleichungen durch Reihen, Journal für die reine und angewandte Mathematik, p.214235, 1873.

J. Gerhard, Modular Algorithms in Symbolic Summation and Symbolic Integration, Lecture Notes in Computer Science, vol.3218, 2004.
DOI : 10.1007/b104035

T. Grégoire, Certied polynomial approximations for D-nite functions, 2012.

A. J. Guttmann, Lattice Green functions and Calabi???Yau differential equations, Journal of Physics A: Mathematical and Theoretical, vol.42, issue.23, p.232001, 2009.
DOI : 10.1088/1751-8113/42/23/232001

S. Hassani, . Ch, J. Koutschan, N. Maillard, and . Zenine, Lattice Green functions: the d-dimensional facecentered cubic lattice, d = 8; 9; 10; 11, Journal of Physics A: Mathematical and Theoretical, vol.12, issue.16, pp.49-2016

L. Heter, Einleitung in die Theorie der linearen Dierentialgleichungen, p.1894

P. Henrici, Applied and Computational Complex Analysis, volume II, 1977.

P. Henrici, Applied and Computational Complex Analysis, volume III, 1986.

E. Hille, Ordinary dierential equations in the complex domain, 1976.

F. Johansson, Computing hypergeometric functions rigorously, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01336266

F. Johansson, Arb, ACM Communications in Computer Algebra, vol.47, issue.3/4, p.12811292, 2017.
DOI : 10.1145/2576802.2576828
URL : https://hal.archives-ouvertes.fr/hal-01394258

M. Kauers, M. Jaroschek, and F. Johansson, Ore Polynomials in Sage, Computer Algebra and Polynomials, p.105125, 2015.
DOI : 10.1007/978-3-319-15081-9_6

M. Kauers and P. Paule, The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates, 2011.

C. Koutschan, Lattice Green functions of the higher-dimensional face-centered cubic lattices, Journal of Physics A: Mathematical and Theoretical, vol.46, issue.12, p.125005, 2013.
DOI : 10.1088/1751-8113/46/12/125005
URL : http://arxiv.org/pdf/1108.2164

M. Mezzarobba, NumGfun, Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ISSAC '10, p.139146
DOI : 10.1145/1837934.1837965
URL : https://hal.archives-ouvertes.fr/inria-00456983

M. Mezzarobba, Autour de l'évaluation numérique des fonctions D-nies, Thèse de doctorat, École polytechnique, 2011.

M. Mezzarobba, Rigorous multiple-precision evaluation of D-Finite functions in SageMath, Extended abstract of a talk at the 5th International Congress on Mathematical Software
URL : https://hal.archives-ouvertes.fr/hal-01342769

M. Mezzarobba and B. Salvy, Eective bounds for P-recursive sequences, Journal of Symbolic Computation, vol.45, issue.10, p.10751096, 2010.
DOI : 10.1016/j.jsc.2010.06.024
URL : https://doi.org/10.1016/j.jsc.2010.06.024

R. E. Moore, Interval arithmetic and automatic error analysis in digital computing, 1962.

M. Team, The MPFR library: algorithms and proofs Available in the MPFR source tree, 2001.

N. S. Nedialkov, K. R. Jackson, and G. F. Corliss, Validated solutions of initial value problems for ordinary dierential equations, Applied Mathematics and Computation, vol.105, issue.1, p.2168, 1999.
DOI : 10.1016/s0096-3003(98)10083-8
URL : http://www.cs.toronto.edu/pub/reports/na/vsode.97.ps.Z

M. Neher, K. R. Jackson, and N. S. Nedialkov, On Taylor Model Based Integration of ODEs, SIAM Journal on Numerical Analysis, vol.45, issue.1, p.236262, 2007.
DOI : 10.1137/050638448
URL : https://publikationen.bibliothek.kit.edu/1000004610/1003614

M. Neher, An enclosure method for the solution of linear ODEs with polynomial coecients. Numerical Functional Analysis and Optimization, p.779803, 1999.
DOI : 10.1080/01630569908816923

M. Neher, Geometric Series Bounds for the Local Errors of Taylor Methods for Linear n-th Order ODEs, Symbolic Algebraic Methods and Verication Methods, p.183193, 2001.
DOI : 10.1007/978-3-7091-6280-4_18

M. Neher, Improved validated bounds for Taylor coecients and for Taylor remainder series, Journal of Computational and Applied Mathematics, vol.152, p.393404, 2003.
DOI : 10.1016/s0377-0427(02)00719-7
URL : https://doi.org/10.1016/s0377-0427(02)00719-7

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 2010.

E. Girard and C. Poole, Introduction to the theory of linear dierential equations, 1936.

R. Rihm, Interval methods for initial value problems in ODEs, Topics in Validated Computations: Proceedings of the IMACS-GAMM International Workshop on Validated Computations, p.173207, 1994.

R. P. Stanley, Enumerative combinatorics, 1999.

J. Van-der-hoeven, Fast evaluation of holonomic functions, Theoretical Computer Science, vol.210, issue.1, 1999.
DOI : 10.1016/S0304-3975(98)00102-9
URL : https://hal.archives-ouvertes.fr/hal-01374898

J. Van-der-hoeven, Fast evaluation of holonomic functions near and in regular singularities, Journal of Symbolic Computation, vol.31, issue.6, p.717743, 2001.

J. Van-der-hoeven, Majorants for formal power series, 2003.

J. Van-der-hoeven, Ecient accelero-summation of holonomic functions, Journal of Symbolic Computation, vol.42, issue.4, p.389428, 2007.

P. G. Warne, D. A. Warne, J. S. Sochacki, G. E. Parker, and D. C. Carothers, Explicit a-priori error bounds and adaptive error control for approximation of nonlinear initial value dierential systems, Computers and Mathematics with Applications, issue.12, p.5216951710, 2006.
DOI : 10.1016/j.camwa.2005.12.004
URL : https://doi.org/10.1016/j.camwa.2005.12.004

M. Stephen and . Watt, ISSAC '10: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, 2010.

N. Zenine, S. Hassani, and J. Maillard, Lattice Green functions: the seven-dimensional face-centred cubic lattice, Journal of Physics A: Mathematical and Theoretical, vol.48, issue.3, pp.48-2015
DOI : 10.1088/1751-8113/48/3/035205