Nesterenko-like rational function, useful to prove the Apéry’s theorem

Anier Soria Lorente
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 2, Pages 79—91
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Authors and affiliations

Anier Soria Lorente
Department of Basic Sciences
Granma University, Cuba


In this paper, a brief introduction to the Apéry’s result and to the so called phenomenon of Apéry’s is given. Here, a modification of the Nesterenko’s rational function, from which new Diophantine approximations to ζ(3) are deduced, is presented. Moreover, as a consequence we deduce the corresponding Apéry-Like recurrence relation as well as a new continued fraction expansion and a new series expansion for ζ(3).


  • Riemann zeta function
  • Apéry’s approximants
  • Recurrence relation
  • Continued fraction expansion
  • Irrationality

AMS Classification

  • Primary: 11B37, 30B70, 14G10, 11J72, 11M06
  • Secondary: 37B20, 11A55, 11J70, 11Y55, 11Y65


  1. Abramov, S. A. Applicability of Zeilberger’s algorithm to hypergeometric terms, Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC’02), New York, 2002, 1–7.
  2. Abramov, S. A., H. Q. Le, A criterion for the applicability of Zeilberger’s algorithm to rational functions, Discrete Math., Vol. 259, 2002, 1–17.
  3. Abramov, S. A. When does Zeilberger’s algorithm succeed?, Appl. Math., Vol. 30, 2003, 424–441.
  4. Abramov, S. A., J. J. Carette, K. O. Geddes, H. Q. Le. Telescoping in the context of symbolic summation in Maple, J. of Symb. Comput., Vol. 30, 2004, 1303–1326.
  5. Apéry, R. Irrationalité de ζ(2) et ζ(3), Astérisque, Vol. 61, 1979, 11–13.
  6. Arvesú, J. Orthogonal forms: A key tool for deducing Apéry’s recurrence relation, J. Approx. Theory, accepted, 2012.
  7. Beukers, F. A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc., Vol. 11, 1979, 268–272.
  8. Beukers, F. Legendre polynomials in irrationality proofs, Bull. Austral. Math. Soc., Vol. 22, 1980, 431–438.
  9. Beuker, F. Padé approximations in number theory, Padé approximation and its applications, (Amsterdam, 1980), 90–99, Lecture Notes in Math., Springer, Berlin-New York, Vol. 888, 1981.
  10. Beuker, F. Consequences of Apéry’s work on ζ(3), preprint of talk presented at the Rencontres Arithmétiques de Caen, ζ(3), Irrationnel: Les Retombées, 1995.
  11. Cohen, H. Demonstration de l’irrationalite de ζ(3) (d’aprés R. Apéry), Séminaire de Théorie des Nombres, Grenoble, 1978, VI.1–VI.9.
  12. Elsner, C. On Prime-Detecting Sequences From Apéry’s Recurrence Formulae for ζ(3) and ζ(2), J. Integer Sequences, Vol. 11, 2008.
  13. Gasper, G., M. Rahman. Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2004.
  14. Gutnik, L. A. On the irrationality of certain quantities involving ζ(3), Acta Arith., Vol. 42, 1983, 255–264.
  15. Jones, W. B., W. J. Thron. Continued fractions, Analytic theory and applications, Encyclopedia Math. Appl. Section: Analysis 11, Addison-Wesley, London, 1980.
  16. Kaneko, M., N. Kurokawa, M. Wakayama. A variation of Euler’s approach to values of the Riemann zeta function, arXiv:math/0206171v2 [math. QA], 31 July 2012.
  17. Koekoek, R., R. F. Swarttouw. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998.
  18. Markoff, A. A. Mémoire sur la transformation des séries peu convergentes en séries tres convergentes, Mémoires de l’Academie Impériale des Sciences de St.-Petersbourg, VII série, t. XXXVII, No. 9, 1890.
  19. Nesterenko, Y. V. A few remarks on ζ(3), Math. Notes, Vol. 59, 1996, No. 6, 625–636.
  20. Nesterenko, Y. V. Integral identities and constructions of approximations to zeta values, J. Théor. Nombres Bordeaux, Vol. 15, 2003, 535–550.
  21. Nikiforov, A. F., V. B. Uvarov. Special Functions in Mathematical Physics, Birkhauser Verlag, Basel, 1988.
  22. Perron, O. Über ein Satz des Herrn Poincaré, J. Reine Angew. Math., Über die Poincarésche lineare Differenzgleichung, Vol. 137, 1910, 6–64.
  23. Poincaré, H., Sur les équations linéaires aux différentielles et aux différences finies, Amer. J. Math., Vol. 7, 1885, 203–258.
  24. Prévost, M. A new proof of the irrationality of ζ(2) and ζ(3) using Padé approximants, J. Comput. Appl. Math., Vol. 67, 1996, 219–235.
  25. Petkovsek, M., H. S. Wilf, D. Zeilberger, A = B, A. K. Peters, Ltd., Wellesley, M.A., 1997.
  26. Sorokin, V. N. Hermite-Padé approximations for Nikishin systems and the irrationality of (3), Communications of the Moscow Math. Soc., Vol. 49, 1993, 176–177.
  27. Sorokin, V. N. Apéry’s theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], No. 3, 1998, 48–52.
  28. Van der Poorten, A. A proof that Euler missed… Apéry’s proof of the irrationality of ζ(3), Math. Intelligencer, Vol. 1, 1978/79, 195–203.
  29. Van Assche, W. Multiple orthogonal polynomials, irrationality and transcendence, Contemporary Mathematics, Vol. 236, 1999, 325–342.

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Cite this paper

Lorente, A. S. (2014). Nesterenko-like rational function, useful to prove the Apéry’s theorem. Notes on Number Theory and Discrete Mathematics, 20(2), 79-91.

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