Simultaneous Padé approximants to the Euler, exponential and logarithmic functions
Journal de Théorie des Nombres de Bordeaux, Volume 27 (2015) no. 2, pp. 565-589.

We present a general method to obtain simultaneous explicit Padé type approximations to the exponential and logarithmic functions.

Nous présentons une méthode générale qui permet d’obtenir des approximations simultanées de type Padé pour les fonctions exponentielles et logarithmes.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.914
Classification: 33C45,  41A21,  33C20,  11J72
Keywords: Padé approximants, Orthogonal polynomials, Hypergeometric series
Tanguy Rivoal 1

1 Institut Fourier CNRS et Université Grenoble 1 100 rue des maths, BP 74 38402 St Martin d’Hères cedex, France
@article{JTNB_2015__27_2_565_0,
     author = {Tanguy Rivoal},
     title = {Simultaneous {Pad\'e} approximants to the {Euler,} exponential and logarithmic functions},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {565--589},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {2},
     year = {2015},
     doi = {10.5802/jtnb.914},
     mrnumber = {3393167},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.914/}
}
TY  - JOUR
TI  - Simultaneous Padé approximants to the Euler, exponential and logarithmic functions
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2015
DA  - 2015///
SP  - 565
EP  - 589
VL  - 27
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.914/
UR  - https://www.ams.org/mathscinet-getitem?mr=3393167
UR  - https://doi.org/10.5802/jtnb.914
DO  - 10.5802/jtnb.914
LA  - en
ID  - JTNB_2015__27_2_565_0
ER  - 
%0 Journal Article
%T Simultaneous Padé approximants to the Euler, exponential and logarithmic functions
%J Journal de Théorie des Nombres de Bordeaux
%D 2015
%P 565-589
%V 27
%N 2
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.914
%R 10.5802/jtnb.914
%G en
%F JTNB_2015__27_2_565_0
Tanguy Rivoal. Simultaneous Padé approximants to the Euler, exponential and logarithmic functions. Journal de Théorie des Nombres de Bordeaux, Volume 27 (2015) no. 2, pp. 565-589. doi : 10.5802/jtnb.914. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.914/

[1] K. Alladi, M. L. Robinson, Legendre polynomials and irrationality, J. Reine Angew. Math. 318 (1980), 137–155. | MR: 579389 | Zbl: 0425.10039

[2] A. I. Aptekarev, A. Branquinho, W. Van Assche, Multiple orthogonal polynomials for classical weights, Trans. Amer. Math. Soc. 355, 10 (2003), 3887–3914. | MR: 1990569 | Zbl: 1033.33002

[3] A. I. Aptekarev (ed.), Rational approximation of Euler’s constant and recurrence relations, Current Problems in Math., 9, Steklov Math. Inst. RAN, (2007). (Russian) | Zbl: 1134.41001

[4] F. Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc. 11, 3 (1979), 268–272. | MR: 554391 | Zbl: 0421.10023

[5] F. Beukers, Legendre polynomials in irrationality proofs, Bull. Austral. Math. Soc. 22, 3 (1980), 431–438. | MR: 601649 | Zbl: 0436.10016

[6] E. Bombieri, J. Mueller, On effective measures of irrationality for a/b r and related numbers, J. Reine Angew. Math. 342 (1983), 173–196. | MR: 703487 | Zbl: 0516.10024

[7] T. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications 13, Gordon and Breach Science Publishers, 1978. | MR: 481884 | Zbl: 0389.33008

[8] G. V. Chudnovsky, Padé approximations to the generalized hypergeometric functions, J. Math. Pures Appl. (9) 58, 4 (1979), 445–476. | MR: 566655 | Zbl: 0434.10023

[9] G. V. Chudnovsky, Formules d’Hermite pour les approximants de Padé de logarithmes et de fonctions binomes, et mesures d’irrationalité, C. R. Acad. Sci. Paris Sér. A-B 288, 21 (1979), 965–967. | MR: 540368 | Zbl: 0418.41011

[10] G. V. Chudnovsky, Approximations de Padé explicites pour les solutions des équations différentielles linéaires fuchsiennes C. R. Acad. Sci. Paris Sér. A-B 290, 3 (1980), 135–137. | MR: 563959 | Zbl: 0424.41014

[11] S. Fischler, T. Rivoal, Approximants de Padé et séries hypergéométriques équilibrées, J. Math. Pures Appl. 82, 10 (2003), 1369–1394. | MR: 2020926 | Zbl: 1064.11053

[12] M. Hata, On the linear independence of the values of polylogarithmic functions, J. Math. Pures Appl. (9) 69, 2 (1990), 133–173. | MR: 1067449 | Zbl: 0712.11040

[13] M. Huttner, Constructible sets of linear differential equations and effective rational approximations of polylogarithmic functions, Israel J. Math. 153 (2006), 1–43. | MR: 2254636 | Zbl: 1143.34057

[14] A. Kuijlaars, K. McLaughlin, Asymptotic zero behavior of Laguerre polynomials with negative parameter, Constr. Approx. 20, 4 (2004), 497–523. | MR: 2078083 | Zbl: 1069.33008

[15] K. Mahler, Applications of some formulae by Hermite to the approximation of exponentials and logarithms, Math. Ann. 168 (1967), 200–227. | MR: 205929 | Zbl: 0144.29201

[16] T. Matala-Aho, Type II Hermite-Padé approximations of generalized hypergeometric series, Constr. Approx. 33, 3 (2011), 289–312. | MR: 2784481 | Zbl: 1236.41017

[17] Yu. V. Nesterenko, Hermite-Padé approximants of generalized hypergeometric functions (Russian) Mat. Sb. 185, 10 (1994), 39–72; translation in Russian Acad. Sci. Sb. Math. 83, 1 (1995), 189–219. | MR: 1309182 | Zbl: 0849.11052

[18] G. Rhin, C. Viola, On a permutation group related to ζ(2), Acta Arith. 77, 1 (1996), 23–56. | MR: 1404975 | Zbl: 0864.11037

[19] T. Rivoal, Rational approximations for values of derivatives of the gamma function, Trans. Amer. Math. Soc. 361, 11 (2009), 6115–6149. | MR: 2529926 | Zbl: 1236.11061

[20] T. Rivoal, On the arithmetic nature of the values of the Gamma function, Euler’s constant and Gompertz’s constant, Michigan Math. J. 61 (2012), 239–254. | MR: 2944478 | Zbl: 1288.11073

[21] E. B. Saff, R. Varga, On the zeros and poles of Padé approximants to e z , III, Numer. Math. 30 (1978), 241–266. | MR: 492295 | Zbl: 0438.41015

[22] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, 1966. | MR: 201688 | Zbl: 0135.28101

[23] V. N. Sorokin, Joint approximations of the square root, logarithm, and arcsine (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2009, 2, 65–69, translation in Moscow Univ. Math. Bull. 64, 2 (2009), 80–83. | MR: 2543176 | Zbl: 1304.11061

[24] A. Thue, Über Annäherungswerte algebraischer Zahlen, J. reine angew. Math. 135 (1909), 284–305. | MR: 1580770

[25] A. Thue, Berechnung aller Lösungen gewisser Gleichungen von der Form ax r -by r =f, Videnskabs-Selskabets Skrifter, I. math.-naturv. KL, 4, 9 S, Christiania 1918.

[26] W. Van Assche, Padé and Hermite-Padé approximation and orthogonality, Surv. Approx. Theory 2 (2006), 61–91. | MR: 2247778 | Zbl: 1102.41017

[27] C. Viola, On Siegel’s method in Diophantine approximation to transcendental numbers, Number theory, II (Rome, 1995). Rend. Sem. Mat. Univ. Politec. Torino 53, 4 (1995), 455–469. | MR: 1452398 | Zbl: 0873.11043

Cited by Sources: