Les $E$-fonctions et $G$-fonctions de Siegel

Rivoal, Tanguy

doi:10.5802/xups.2019-03

Les

E

-fonctions et

G

-fonctions de Siegel

Rivoal, Tanguy ¹

¹ Institut Fourier, CNRS et Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France

Journées mathématiques X-UPS, Périodes et transcendance (2019), pp. 197-298.

Résumé

Siegel a introduit en 1929 les $E$ - et $G$ -fonctions, deux classes de séries entières à coefficients algébriques qui sont solutions d’équations différentielles à coefficients polynomiaux dans le but de généraliser les théorèmes classiques d’Hermite, Lindemann et Weierstrass concernant la nature arithmétique des valeurs des fonctions exponentielle et logarithme aux points algébriques, que ces deux classes généralisent respectivement.

Une première partie introduit la théorie des approximants de Padé « explicites » et la combine avec des méthodes « inexplicites » pour aboutir au célèbre résultat (1956) de Siegel et Shidlovskii sur la nature arithmétique des valeurs de $E$ -fonctions. On indique ensuite comment Chudnovsky a complété en 1984 le programme de Siegel sur la nature diophantienne des $G$ -fonctions. Une deuxième partie aborde la nature des équations différentielles satisfaites par les $G$ -fonctions (travaux d’André, Chudnovsky et Katz) puis, par ricochet, par les $E$ -fonctions. On conclut par de nouvelles applications arithmétiques que l’on en déduit et qui complètent le théorème de Siegel-Shidlovskii (travaux d’André, Beukers, Adamczewski-Rivoal).

Publié le : 2024-08-01

DOI : 10.5802/xups.2019-03

Affiliations des auteurs :

Rivoal, Tanguy ¹

¹ Institut Fourier, CNRS et Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France

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Rivoal, Tanguy. Les $E$-fonctions et $G$-fonctions de Siegel. Journées mathématiques X-UPS, Périodes et transcendance (2019), pp. 197-298. doi : 10.5802/xups.2019-03. http://archive.numdam.org/articles/10.5802/xups.2019-03/

Bibliographie
Cité par

[1] Handbook of mathematical functions with formulas, graphs, and mathematical tables (Abramowitz, Milton; Stegun, Irene A., eds.), Dover Publications, Inc., New York, 1992 | MR

[2] Adamczewski, B.; Rivoal, T. Exceptional values of $E$ -functions at algebraic points, Bull. London Math. Soc., Volume 50 (2018) no. 4, pp. 697-708 | DOI | MR | Zbl

[3] Alladi, K.; Robinson, M. L. Legendre polynomials and irrationality, J. reine angew. Math., Volume 318 (1980), pp. 137-155 | MR | Zbl

[4] André, Y. $G$ -functions and geometry, Aspects of Mathematics, E13, Friedr. Vieweg & Sohn, Braunschweig, 1989 | DOI | MR

[5] André, Y. $G$ -fonctions et transcendance, J. reine angew. Math., Volume 476 (1996), pp. 95-125 | DOI | MR | Zbl

[6] André, Y. Séries Gevrey de type arithmétique. I. Théorèmes de pureté et de dualité, Ann. of Math. (2), Volume 151 (2000) no. 2, pp. 705-740 | DOI | MR | Zbl

[7] André, Y. Séries Gevrey de type arithmétique. II. Transcendance sans transcendance, Ann. of Math. (2), Volume 151 (2000) no. 2, pp. 741-756 | DOI | MR | Zbl

[8] André, Y. Arithmetic Gevrey series and transcendence. A survey, J. Théor. Nombres Bordeaux, Volume 15 (2003) no. 1, pp. 1-10 Les XXIIèmes Journées Arithmetiques (Lille, 2001) | DOI | Numdam | MR | Zbl

[9] André, Y. Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas & Synthèses, 17, Société Mathématique de France, Paris, 2004 | MR

[10] André, Y. Solution algebras of differential equations and quasi-homogeneous varieties : a new differential Galois correspondence, Ann. Sci. École Norm. Sup. (4), Volume 47 (2014) no. 2, pp. 449-467 | DOI | MR | Zbl

[11] Apéry, R. Irrationalité de $ζ 2$ et $ζ 3$ , Journées arithmétiques de Luminy (Astérisque), Volume 61, Société Mathématique de France, Paris, 1979, pp. 11-13 | MR | Zbl

[12] Apéry, R. Interpolation de fractions continues et irrationalité de certaines constantes, Mathematics (CTHS : Bull. Sec. Sci., III), Bibliothèque Nationale, Paris, 1981, pp. 37-53 | MR | Zbl

[13] Baker, A. Transcendental number theory, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990 | MR

[14] Ball, K.; Rivoal, T. Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math., Volume 146 (2001) no. 1, pp. 193-207 | DOI | MR | Zbl

[15] Bauer, M.; Bennett, M. A. Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation, Ramanujan J., Volume 6 (2002) no. 2, pp. 209-270 | DOI | MR | Zbl

[16] Bertrand, D. On André’s proof of the Siegel-Shidlovsky theorem, Colloque Franco-Japonais : Théorie des Nombres Transcendants (Tokyo, 1998) (Sem. Math. Sci.), Volume 27, Keio University, Yokohama, 1999, pp. 51-63 | MR

[17] Bertrand, D.; Beukers, F. Équations différentielles linéaires et majorations de multiplicités, Ann. Sci. École Norm. Sup. (4), Volume 18 (1985) no. 1, pp. 181-192 | DOI | Numdam | MR | Zbl

[18] Bertrand, D.; Chirskii, V.; Yebbou, J. Effective estimates for global relations on Euler-type series, Ann. Fac. Sci. Toulouse Math. (6), Volume 13 (2004) no. 2, pp. 241-260 http://afst.centre-mersenne.org/item?id=AFST_2004_6_13_2_241_0 | DOI | Numdam | MR | Zbl

[19] Beukers, F. On the generalized Ramanujan-Nagell equation. I, Acta Arith., Volume 38 (1980/81) no. 4, pp. 389-410 | DOI | MR | Zbl

[20] Beukers, F. Padé-approximations in number theory, Padé approximation and its applications (Amsterdam, 1980) (Lect. Notes in Math.), Volume 888, Springer, Berlin-New York, 1981, pp. 90-99 | MR | Zbl

[21] Beukers, F. Algebraic values of $G$ -functions, J. reine angew. Math., Volume 434 (1993), pp. 45-65 | DOI | MR | Zbl

[22] Beukers, F. A rational approach to $π$ , Nieuw Arch. Wisk., Volume 1 (2000) no. 4, pp. 372-379 | MR | Zbl

[23] Beukers, F. A refined version of the Siegel-Shidlovskii theorem, Ann. of Math. (2), Volume 163 (2006) no. 1, pp. 369-379 | DOI | MR | Zbl

[24] Beukers, F. $E$ -functions and $G$ -functions (2008) (disponible à http://swc.math.arizona.edu/aws/2008/08BeukersNotesDraft.pdf)

[25] Beukers, F. Algebraic $A$ -hypergeometric functions, Invent. Math., Volume 180 (2010) no. 3, pp. 589-610 | DOI | MR | Zbl

[26] Beukers, F.; Heckman, G. Monodromy for the hypergeometric function $_{n} F_{n - 1}$ , Invent. Math., Volume 95 (1989) no. 2, pp. 325-354 | DOI | MR | Zbl

[27] Beukers, F.; Peters, C. A. M. A family of $K 3$ surfaces and $ζ (3)$ , J. reine angew. Math., Volume 351 (1984), pp. 42-54 | DOI | MR | Zbl

[28] Bombieri, E. On $G$ -functions, Recent progress in analytic number theory, Vol. 2 (Durham, 1979), Academic Press, London-New York, 1981, pp. 1-67 | MR | Zbl

[29] Bostan, A. Calcul formel pour la combinatoire des marches, Mémoire d’habilitation à diriger des recherches, Univ. Paris 13 (2017) (tel-01660300)

[30] Bostan, A.; Kauers, M. The complete generating function for Gessel walks is algebraic, Proc. Amer. Math. Soc., Volume 138 (2010) no. 9, pp. 3063-3078 | DOI | MR | Zbl

[31] Bostan, A.; Lairez, P.; Salvy, B. Multiple binomial sums, J. Symbolic Comput., Volume 80 (2017) no. part 2, pp. 351-386 | DOI | MR | Zbl

[32] Bostan, A.; Rivoal, T.; Salvy, B. Explicit degree bounds for right factors of linear differential operators, Bull. Lond. Math. Soc., Volume 53 (2021) no. 1, pp. 53-62 | DOI | MR | Zbl

[33] Bostan, A.; Rivoal, T.; Salvy, B. Minimization of differential equations and algebraic values of $E$ -functions, Math. Comp., Volume 93 (2024) no. 347, pp. 1427-1472 | DOI | MR | Zbl

[34] Brezinski, C. Padé-type approximation and general orthogonal polynomials, International Series of Numerical Math., 50, Birkhäuser Verlag, Basel-Boston, Mass., 1980, 250 pages | DOI | MR

[35] Broadhurst, D. Feynman integrals, L-series and Kloosterman moments, Commun. Number Theory Phys., Volume 10 (2016) no. 3, pp. 527-569 | DOI | MR | Zbl

[36] Christol, G. Fonctions hypergéométriques bornées, Groupe de travail d’analyse ultramétrique, Volume 14, Université de Paris VII, 1986/87 (Exp. 8)

[37] Chudnovsky, D. V.; Chudnovsky, G. V. Applications of Padé approximations to Diophantine inequalities in values of $G$ -functions, Number theory (New York, 1983–84) (Lect. Notes in Math.), Volume 1135, Springer, Berlin, 1985, pp. 9-51 | DOI | MR | Zbl

[38] Chudnovsky, G. V. Algebraic independence of the values of elliptic function at algebraic points. Elliptic analogue of the Lindemann-Weierstrass theorem, Invent. Math., Volume 61 (1980) no. 3, pp. 267-290 | DOI | MR | Zbl

[39] Chudnovsky, G. V. On applications of Diophantine approximations, Proc. Nat. Acad. Sci. U.S.A., Volume 81 (1984) no. 22, pp. 7261-7265 | DOI | MR | Zbl

[40] Cohn, H. A short proof of the simple continued fraction expansion of $e$ , Amer. Math. Monthly, Volume 113 (2006) no. 1, pp. 57-62 | DOI | MR | Zbl

[41] Comtet, L. Calcul pratique des coefficients de Taylor d’une fonction algébrique, Enseign. Math. (2), Volume 10 (1964), pp. 267-270 | Zbl

[42] Conway, J. B. Functions of one complex variable, Graduate Texts in Math., 11, Springer-Verlag, New York-Berlin, 1978 | DOI | MR

[43] Davis, C. S. Rational approximations to $e$ , J. Austral. Math. Soc. Ser. A, Volume 25 (1978) no. 4, pp. 497-502 | DOI | MR | Zbl

[44] Dèbes, P. $G$ -fonctions et théorème d’irreductibilité de Hilbert, Acta Arith., Volume 47 (1986) no. 4, pp. 371-402 | DOI | MR | Zbl

[45] Delaygue, E. Critère pour l’intégralité des coefficients de Taylor des applications miroir, J. reine angew. Math., Volume 662 (2012), pp. 205-252 | DOI | MR | Zbl

[46] Delaygue, E.; Rivoal, T.; Roques, J. On Dwork’s $p$ -adic formal congruences theorem and hypergeometric mirror maps, Mem. Amer. Math. Soc., 246, no. 1163, American Mathematical Society, Providence, RI, 2017 | DOI | MR

[47] Descombes, R. Éléments de théorie des nombres, Mathématiques, Presses Universitaires de France, Paris, 1986, 279 pages | MR

[48] Di Vizio, L. Sur la théorie géométrique des $G$ -fonctions. Le théorème de Chudnovsky à plusieurs variables, Math. Ann., Volume 319 (2001) no. 1, pp. 181-213 | DOI | MR | Zbl

[49] Dwork, B. On Apéry’s differential operator, Groupe de travail d’analyse ultramétrique, Volume 7-8, Université de Paris VII, 1979/81 (Exp. No. 25)

[50] Dwork, B. Differential operators with nilpotent $p$ -curvature, Amer. J. Math., Volume 112 (1990) no. 5, pp. 749-786 | DOI | MR | Zbl

[51] Dwork, B.; Gerotto, G.; Sullivan, F. J. An introduction to $G$ -functions, Annals of Math. Studies, 133, Princeton University Press, Princeton, NJ, 1994 | MR

[52] Eisenstein, G. Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Funktionen, Bericht Königl. Preuss Akad. d. Wiss. zu Berlin (1852), pp. 411-444

[53] Finch, S. R. Mathematical constants, Encyclopedia of Math. and its Applications, 94, Cambridge University Press, Cambridge, 2003 | MR

[54] Fischler, S. Irrationalité de valeurs de zêta (d’après Apéry, Rivoal,...), Séminaire Bourbaki (Astérisque), Volume 294, Société Mathématique de France, Paris, 2004, pp. 27-62 | Zbl

[55] Fischler, S. Nesterenko’s linear independence criterion for vectors, Monatsh. Math., Volume 177 (2015) no. 3, pp. 397-419 | DOI | MR | Zbl

[56] Fischler, S. Linear independence of odd zeta values using Siegel’s lemma, 2021 | arXiv

[57] Fischler, S.; Rivoal, T. Approximants de Padé et séries hypergéométriques équilibrées, J. Math. Pures Appl. (9), Volume 82 (2003) no. 10, pp. 1369-1394 | DOI | MR | Zbl

[58] Fischler, S.; Rivoal, T. On the values of $G$ -functions, Comment. Math. Helv., Volume 89 (2014) no. 2, pp. 313-341 | DOI | MR

[59] Fischler, S.; Rivoal, T. Arithmetic theory of $E$ -operators, J. Éc. polytech. Math., Volume 3 (2016), pp. 31-65 | DOI | Numdam | MR

[60] Fischler, S.; Rivoal, T. On the denominators of the Taylor coefficients of $G$ -functions, Kyushu J. Math., Volume 71 (2017) no. 2, pp. 287-298 | DOI | MR | Zbl

[61] Fischler, S.; Rivoal, T. Microsolutions of differential operators and values of arithmetic Gevrey series, Amer. J. Math., Volume 140 (2018) no. 2, pp. 317-348 | DOI | MR | Zbl

[62] Fischler, S.; Rivoal, T. Rational approximation to values of $G$ -functions, and their expansions in integer bases, Manuscripta Math., Volume 155 (2018) no. 3-4, pp. 579-595 (Erratum : Ibid. p. 597–598) | DOI | MR | Zbl

[63] Fischler, S.; Rivoal, T. Linear independence of values of $G$ -functions, J. Eur. Math. Soc. (JEMS), Volume 22 (2020) no. 5, pp. 1531-1576 | DOI

[64] Fischler, S.; Rivoal, T. Linear independence of values of $G$ -functions, II : outside the disk of convergence, Ann. Math. Qué., Volume 45 (2021) no. 1, pp. 53-93 | DOI | Zbl

[65] Fischler, S.; Rivoal, T. On Siegel’s problem for $E$ -functions, Rend. Semin. Mat. Univ. Padova, Volume 148 (2022), pp. 83-115 | DOI | Numdam | MR

[66] Fischler, S.; Rivoal, T. Effective algebraic independence of values of $E$ -functions, Math. Z., Volume 305 (2023) no. 3, 48, 17 pages | DOI | MR | Zbl

[67] Fischler, S.; Sprang, J.; Zudilin, W. Many values of the Riemann zeta function at odd integers are irrational, C. R. Math. Acad. Sci. Paris, Volume 356 (2018) no. 7, pp. 707-711 | DOI | Numdam | MR | Zbl

[68] Fischler, S.; Sprang, J.; Zudilin, W. Many odd zeta values are irrational, Compositio Math., Volume 155 (2019) no. 5, pp. 938-952 | DOI | MR | Zbl

[69] Fresán, J.; Jossen, P. A non-hypergeometric $E$ -function, Ann. of Math. (2), Volume 194 (2021) no. 3, pp. 903-942 | Zbl

[70] Frobenius, G. Über die Integration der linearen Differentialgleichungen durch Reihen, J. reine angew. Math., Volume 76 (1873), pp. 214-235 | Zbl

[71] Galočkin, A. I. Lower bounds of polynomials in the values of a certain class of analytic functions, Mat. Sb. (N.S.), Volume 95 (1974), pp. 396-417 | MR

[72] Galočkin, A. I. The arithmetic properties of the values of some entire hypergeometric functions, Sibirsk. Mat. Zh., Volume 17 (1976) no. 6, pp. 1220-1235 | MR

[73] Galočkin, A. I. Criterion for membership of hypergeometric Siegel functions in a class of $E$ -functions, Mat. Zametki, Volume 29 (1981) no. 1, pp. 3-14 | MR | Zbl

[74] Gel’fond, A. Sur les nombres transcendants, C. R. Acad. Sci. Paris, Volume 189 (1929), pp. 1224-1226 | Zbl

[75] Gel’fond, A. Sur le 7ème problème de Hilbert, Bull. Acad. Sci. URSS, Volume 7 (1934), pp. 623-640

[76] Glasser, M. L.; Zucker, I. J. Extended Watson integrals for the cubic lattices, Proc. Nat. Acad. Sci. U.S.A., Volume 74 (1977) no. 5, pp. 1800-1801 | DOI | MR | Zbl

[77] Gorelov, V. A. On the Siegel conjecture for the case of second-order linear homogeneous differential equations, Mat. Zametki, Volume 75 (2004) no. 4, pp. 549-565 | MR

[78] Gorelov, V. A. On the structure of the set of $E$ -functions satisfying second-order linear differential equations, Mat. Zametki, Volume 78 (2005) no. 3, pp. 331-348 | DOI | MR

[79] Griffiths, P. A. Periods of integrals on algebraic manifolds : summary and discussion of open problems, Bull. Amer. Math. Soc., Volume 76 (1970), pp. 228-296 | DOI | Zbl

[80] Grigor’ev, D. Yu. Complexity of factoring and calculating the GCD of linear ordinary differential operators, J. Symbolic Comput., Volume 10 (1990) no. 1, pp. 7-37 | DOI | MR | Zbl

[81] Grinspan, P. Approximation et indépendance algébrique de quasi-périodes de variétés abéliennes, thèse de doctorat, Université Paris 6 (2000) (tel-00001328)

[82] Harris, W. A. Jr.; Sibuya, Y. The reciprocals of solutions of linear ordinary differential equations, Adv. in Math., Volume 58 (1985) no. 2, pp. 119-132 | DOI | MR | Zbl

[83] Hata, M. On the linear independence of the values of polylogarithmic functions, J. Math. Pures Appl. (9), Volume 69 (1990) no. 2, pp. 133-173 | MR | Zbl

[84] Hata, M. Rational approximations to $π$ and some other numbers, Acta Arith., Volume 63 (1993) no. 4, pp. 335-349 | DOI | MR | Zbl

[85] Hermite, C. Sur la fonction exponentielle, C. R. Acad. Sci. Paris, Volume 77 (1873), p. 18-24, 74–79, 226–233 et 285–293 (disponible à http://www.bibnum.education.fr/mathematiques/theorie-des-nombres/la-demonstration-de-la-transcendance-de-e) | Zbl

[86] Hilbert, D. Sur les problèmes futurs des mathématiques, Gauthier-Villars, Paris, 1902 (réédition Jacques Gabay, 1990)

[87] Hille, E. Ordinary differential equations in the complex domain, Dover Publications, Inc., Mineola, NY, 1997 (Reprint of the 1976 original) | MR

[88] Huber, A.; Müller-Stach, S. Periods and Nori motives, Ergeb. Math. Grenzgeb. (3), 65, Springer, Cham, 2017, xxiii+372 pages | DOI | MR

[89] Ince, E. L. Ordinary Differential Equations, Dover Publications, New York, 1944 (disponible à https://archive.org/details/ordinarydifferen029666mbp)

[90] Joyce, G. S.; Zucker, I. J. Special values of the hypergeometric series. III, Math. Proc. Cambridge Philos. Soc., Volume 133 (2002) no. 2, pp. 213-222 | DOI | MR | Zbl

[91] Katz, N. Nilpotent connections and the monodromy theorem : applications of a result of Turrittin, Publ. Math. Inst. Hautes Études Sci., Volume 39 (1970), pp. 175-232 | DOI | Numdam | Zbl

[92] Kintchine, A. Ya. Continued fractions, Dover Publications, 1935 (trad. anglaise : 1964, rééditions 1992, 1997)

[93] Koksma, J. F.; Popken, J. Zur Transzendenz von $e^{π}$ , J. reine angew. Math., Volume 168 (1932), pp. 211-230 | Zbl

[94] Kontsevich, M.; Zagier, D. Periods, Mathematics unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 771-808 | DOI | MR | Zbl

[95] Krammer, D. An example of an arithmetic Fuchsian group, J. reine angew. Math., Volume 473 (1996), pp. 69-85 | Zbl

[96] Kuzmin, R Sur une nouvelle classe de nombres transcendants, Izv. Akad. Nauk SSSR Ser. Mat., Volume 7 (1930) no. 6, pp. 585-597 (en russe) | Zbl

[97] Lairez, P. Computing periods of rational integrals, Math. Comp., Volume 85 (2016) no. 300, pp. 1719-1752 | DOI | MR | Zbl

[98] Lambert, J.-H. Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques, Mémoires de l’Académie royale des sciences de Berlin, Volume 17 (1861), pp. 265-322

[99] Landau, E. Sur les conditions de divisibilité d’un produit de factorielles par un autre, Collected works, I, Thales-Verlag, 1985, p. 116

[100] Le Lionnais, F. Les nombres remarquables, Actualités scientifiques et industrielles, 1407, Hermann, Paris, 1983

[101] Legendre, A-M. Éléments de géométrie, 1795 (Note IV)

[102] Lepetit, G. Le théorème d’André-Chudnovsky-Katz, 2018 (mémoire de master recherche, Institut Fourier, Grenoble) | arXiv

[103] Lepetit, G. Le théorème d’André-Chudnovsky-Katz ‘au sens large’, North-West. Eur. J. Math., Volume 7 (2021), pp. 83-149 | Zbl

[104] Lewin, L. Dilogarithms and associated functions, Macdonald, London, 1958 | MR

[105] Lindemann, F. Ueber die Zahl $π$ , Math. Ann., Volume 20 (1882), pp. 213-225 | DOI

[106] Liouville, J. Sur des classes très étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationnelles algébriques, J. Math. Pures Appl., Volume 16 (1851), pp. 133-142 | Numdam

[107] Mahler, K. Zur Approximation der Exponentialfunktion und des Logarithmus. Teil I, J. reine angew. Math., Volume 166 (1931), pp. 118-136 | Zbl

[108] Mahler, K. On the approximation of $π$ , Indag. Math., Volume 15 (1953), pp. 30-42 | DOI | MR | Zbl

[109] Mahler, K. Applications of a theorem by A. B. Shidlovski, Proc. Roy. Soc. Ser. A, Volume 305 (1968), pp. 149-173 | DOI | MR | Zbl

[110] Malgrange, B. Équations différentielles linéaires et transformation de Fourier. Une introduction., Ensaios Matemáticos, 1, Sociedade Brasileira de Matemática, Rio de Janeiro, 1989

[111] Marcovecchio, R. Linear independence of linear forms in polylogarithms, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume 5 (2006) no. 1, pp. 1-11 | Numdam | MR | Zbl

[112] Marcovecchio, R. The Rhin-Viola method for $log 2$ , Acta Arith., Volume 139 (2009) no. 2, pp. 147-184 | DOI | MR | Zbl

[113] McKean, H.; Moll, V. Elliptic curves. Function theory, geometry, arithmetic, Cambridge University Press, Cambridge, 1997 | DOI | MR

[114] Nagata, M. Rational approximations to linear forms in values of $G$ -functions, Acta Arith., Volume 70 (1995) no. 4, pp. 313-341 | DOI | MR | Zbl

[115] Nagata, M. Diophantine approximations related to rational values of $G$ -functions, Acta Arith., Volume 106 (2003) no. 4, pp. 311-344 | DOI | MR | Zbl

[116] Nesterenko, Yu. V. Linear independence of numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1985) no. 1, pp. 46-49 | MR | Zbl

[117] Nesterenko, Yu. V. Modular functions and transcendence questions, Mat. Sb., Volume 187 (1996) no. 9, pp. 65-96 | DOI | MR

[118] Nesterenko, Yu. V. Some remarks on $ζ (3)$ , Mat. Zametki, Volume 59 (1996) no. 6, pp. 865-880 | DOI | MR

[119] Nesterenko, Yu. V.; Shidlovskiĭ, A. B. On the linear independence of values of $E$ -functions, Mat. Sb., Volume 187 (1996) no. 8, pp. 93-108 | DOI | MR

[120] Nikišin, E. M. Irrationality of values of functions $F (x, s)$ , Mat. Sb. (N.S.), Volume 109(151) (1979) no. 3, pp. 410-417 | MR | Zbl

[121] Niven, I. A simple proof that $π$ is irrational, Bull. Amer. Math. Soc., Volume 53 (1947), p. 509 | DOI | MR | Zbl

[122] Niven, I. Irrational numbers, The Carus Mathematical Monographs, 11, The Mathematical Association of America & John Wiley and Sons, Inc., New York, N.Y., 1967 | MR

[123] Osgood, C. F. Nearly perfect systems and effective generalizations of Shidlovski’s theorem, J. Number Theory, Volume 13 (1981) no. 4, pp. 515-540 | DOI | MR | Zbl

[124] Padé, H. Sur la représentation approchée d’une fonction par des fractions rationnelles, Gauthier-Villars et fils, Paris, 1892 (Thèse de doctorat de la Faculté des sciences)

[125] Padé, H. Œuvres, Librairie Albert Blanchard, Paris, 1984 (rassemblées et présentées par Claude Brezinski)

[126] Pellarin, F.; Rivoal, T.; Weil, J.-A. Some remarks on classical modular forms and hypergeometric series (2019) (http://rivoal.perso.math.cnrs.fr/articles/modhyp.pdf)

[127] Perron, O. Über lineare Differentialgleichungen mit rationalen Koeffizienten, Acta Math., Volume 34 (1910), pp. 139-163 | DOI | Zbl

[128] Petkovšek, M.; Wilf, H. S.; Zeilberger, D. $A = B$ , A K Peters, Ltd., Wellesley, MA, 1996 (http://www.math.rutgers.edu/~zeilberg/AeqB.pdf)

[129] Poole, E. G. C. Introduction to the theory of linear differential equations, Dover Publications, Inc., New York, 1960 | MR

[130] van der Poorten, A. A proof that Euler missed... Apéry’s proof of the irrationality of $ζ (3)$ , Math. Intelligencer, Volume 1 (1978/79) no. 4, pp. 195-203 | DOI | MR | Zbl

[131] Ramanujan, S. Modular equations and approximations to $π$ , Quart. J. Math. Oxford Ser., Volume 45 (1914), pp. 350-372 | Zbl

[132] Reyssat, É. Mesures de transcendance pour les logarithmes de nombres rationnels, Diophantine approximations and transcendental numbers (Luminy, 1982) (Progress in Math.), Volume 31, Birkhäuser Boston, Boston, MA, 1983, pp. 235-245 | DOI | MR | Zbl

[133] Rhin, G.; Viola, C. The group structure for $ζ (3)$ , Acta Arith., Volume 97 (2001) no. 3, pp. 269-293 | DOI | MR

[134] Rivoal, T. La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C. R. Acad. Sci. Paris Sér. I Math., Volume 331 (2000) no. 4, pp. 267-270 | DOI | MR | Zbl

[135] Rivoal, T. Indépendance linéaire des valeurs des polylogarithmes, J. Théor. Nombres Bordeaux, Volume 15 (2003) no. 2, pp. 551-559 | DOI | Numdam | MR | Zbl

[136] Rivoal, T. Convergents and irrationality measures of logarithms, Rev. Mat. Iberoamericana, Volume 23 (2007) no. 3, pp. 931-952 | DOI | MR | Zbl

[137] Rivoal, T. On the arithmetic nature of the values of the gamma function, Euler’s constant, and Gompertz’s constant, Michigan Math. J., Volume 61 (2012) no. 2, pp. 239-254 | DOI | MR | Zbl

[138] Rivoal, T. Remarks on the impossibility of a Siegel-Shidlovskii like theorem for G-functions, Hardy-Ramanujan J., Volume 38 (2015), pp. 29-35 | MR | Zbl

[139] Rivoal, T. Valeurs algébriques de $E$ -fonctions aux points algébriques (2016) (hal-03676576)

[140] Rivoal, T. Is Euler’s constant a value of an arithmetic special function ? (2017) (hal-01619235)

[141] Rivoal, T. Factors of $E$ -operators with an $η$ -apparent singularity at zero, J. Math. Soc. Japan, Volume 74 (2022) no. 3, pp. 719-733 | DOI | MR | Zbl

[142] Rivoal, T. On Galochkin’s characterization of hypergeometric $G$ -functions, Mosc. J. Comb. Number Theory, Volume 11 (2022) no. 1, pp. 11-19 | DOI | MR | Zbl

[143] Rivoal, T.; Roques, J. $E$ -functions of order 2 and units of $E$ -values, 2015 (http://rivoal.perso.math.cnrs.fr/articles/unitse.pdf. Une partie des résultats de cette prépublication est parue dans [144])

[144] Rivoal, T.; Roques, J. On the algebraic dependence of $E$ -functions, Bull. London Math. Soc., Volume 48 (2016) no. 2, pp. 271-279 | DOI | Zbl

[145] Rivoal, T.; Roques, J. Siegel’s problem for $E$ -functions of order $2$ (2016) (prépublication)

[146] Rivoal, T.; Roques, J. Holomorphic solutions of $E$ -operators, Israel J. Math., Volume 220 (2017) no. 1, pp. 275-282 | DOI | MR | Zbl

[147] Rodriguez Villegas, F. Integral ratios of factorials and algebraic hypergeometric functions, 2007 | arXiv

[148] Roth, K. F. Rational approximations to algebraic numbers, Mathematika, Volume 2 (1955), pp. 1-20 (Corrigendum : Ibid., p. 168) | DOI | Zbl

[149] Rukhadze, E. A. A lower bound for the approximation of $\ln 2$ by rational numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1987) no. 6, p. 25-29, 97 | MR | Zbl

[150] Salikhov, V. Kh. On the irrationality measure of $ln 3$ , Dokl. Akad. Nauk, Volume 417 (2007) no. 6, pp. 753-755 | DOI | MR

[151] Salikhov, V. Kh. On the measure of irrationality of $π$ , Mat. Zametki, Volume 88 (2010) no. 4, pp. 583-593 | DOI | MR | Zbl

[152] Sandifer, E. How Euler did it, MAA Online, http://eulerarchive.maa.org/hedi/, 2006

[153] Schneider, T. Transzendenzuntersuchungen periodischer Funktionen I. Transzendenz von Potenzen, J. reine angew. Math., Volume 172 (1935), pp. 65-69 | DOI | MR | Zbl

[154] Schneider, T. Introduction aux nombres transcendants, Gauthier-Villars, Paris, 1959 | MR

[155] Shidlovskii, A. B. A criterion for algebraic independence of the values of a class of entire functions, Izv. Akad. Nauk SSSR Ser. Mat., Volume 23 (1959), pp. 35-66 | MR | Zbl

[156] Shidlovskii, A. B. Algebraic independence of the values of certain hypergeometric $E$ -functions, Trudy Moskov. Mat. Obšč., Volume 18 (1968), pp. 55-64 | MR | Zbl

[157] Shidlovskii, A. B. Transcendental numbers, De Gruyter Studies in Math., 12, Walter de Gruyter & Co., Berlin, 1989 | DOI | MR

[158] Siegel, C. L. Über einige Anwendungen diophantischer Approximationen, 1, S. Abhandlungen Akad., Berlin, 1929

[159] Siegel, C. L. Transcendental numbers, Annals of Math. Studies, 16, Princeton University Press, Princeton, NJ, 1949 | MR

[160] Siegel, C. L. Über einige Anwendungen diophantischer Approximationen, On some applications of Diophantine approximations (Quad./Monogr.), Volume 2, Scuola Normale, Pisa & Springer, 2014 (Avec un commentaire et l’article « Integral points on curves : Siegel’s theorem after Siegel’s proof » par C. Fuchs and U. Zannier) | MR

[161] Slater, L. J. Generalized hypergeometric functions, Cambridge University Press, Cambridge, 2008 | MR

[162] de Stainville, J. Mélanges d’analyse algébrique et de géométrie, http://www.bibnum.education.fr/mathematiques/theorie-des-nombres/melange-d-analyse-algebrique-et-de-geometrie, 1815

[163] Stiller, P. F. Classical automorphic forms and hypergeometric functions, J. Number Theory, Volume 28 (1988) no. 2, pp. 219-232 | DOI | MR | Zbl

[164] Waldschmidt, M. Une mesure de transcendance de $e^{π}$ , Séminaire Delange-Pisot-Poitou, 1975/76, Volume 17, no. 2, Secrétariat Math., Paris, 1977 (Exp. No. G4) | MR

[165] Wolfart, J. Werte hypergeometrischer Funktionen, Invent. Math., Volume 92 (1988) no. 1, pp. 187-216 | DOI | MR | Zbl

[166] Zeilberger, D.; Zudilin, W. The irrationality measure of $π$ is at most 7.103205334137..., Mosc. J. Comb. Number Theory, Volume 9 (2020) no. 4, pp. 407-419 | DOI | Zbl

[167] Zudilin, W. On rational approximations of values of a class of entire functions, Mat. Sb., Volume 186 (1995) no. 4, pp. 89-124 | DOI | MR

[168] Zudilin, W. On the irrationality measure of values of $G$ -functions, Izv. Ross. Akad. Nauk Ser. Mat., Volume 60 (1996) no. 1, pp. 87-114 | DOI | MR

[169] Zudilin, W. One of the numbers $ζ (5)$ , $ζ (7)$ , $ζ (9)$ , $ζ (11)$ is irrational, Uspekhi Mat. Nauk, Volume 56 (2001) no. 4(340), pp. 149-150 | DOI | MR

[170] Zudilin, W. Two hypergeometric tales and a new irrationality measure of $ζ (2)$ , Ann. Math. Québec, Volume 38 (2014) no. 1, pp. 101-117 | DOI | MR | Zbl

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