On Zudilin-like rational approximations to ζ (5)

Anier Soria Lorente
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 2, Pages 104—116
DOI: 10.7546/nntdm.2018.24.2.104-116
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Authors and affiliations

Anier Soria Lorente
Department of Mathematics, University of Granma
Bayamo, Granma, Cuba

Abstract

In this paper we obtain two Zudilin-Like recurrence relations of third order for ζ (5), after applying Zeilberger’s algorithm of creative telescoping to some hypergeometric series. These recurrence relations do not supply diophantine approximations to ζ (5) that prove its irrationality, however it presents an algorithm for fast calculation of this constant. Moreover, we deduce a new continued fraction expansion for ζ (5) as a consequence.

Keywords

  • Riemann zeta function
  • Recurrence relation
  • Continued fraction expansion
  • Irrationality

2010 Mathematics Subject Classification

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

References

  1. Abramov, S. A. (2002) Applicability of Zeilberger’s algorithm to hypergeometric terms, In ISSAC’02: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, New York, 1–7.
  2. Abramov, S. A. & Le, H. Q. (2002) A criterion for the applicability of Zeilberger’s algorithm to rational functions, Discrete Math., 259, 1–17.
  3. Abramov, S. A. (2003) When does Zeilberger’s algorithm succeed?, Appl. Math., 30, 424– 441, 2003.
  4. Abramov, S. A., Carette, J. J., Geddes, K. O. & Le, H.Q. (2004) Telescoping in the context of symbolic summation in Maple, J. of Symb. Comput., 30, 1303–1326.
  5. Abramowitz, M. & Stegun, I. A. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York.
  6. Almkvist, G. & Granville, A. (1999) Borwein and Bradley’s Apéry-like formulae for ζ (4n + 3), Experiment. Math., 8, 197–203.
  7. Apéry, R. (1979) Irrationalité de ζ (2) et ζ (3), Astérisque, 61, 11–13.
  8. Arvesú, J. (2012) Orthogonal forms: A key tool for deducing Apéry’s recurrence relation, J. Approx. Theory, accepted, 2012.
  9. Beukers, F. (1979) A note on the irrationality of ζ (2) and ζ (3), Bull. London Math. Soc., 11, 268–272.
  10. Beukers, F. (1980) Legendre polynomials in irrationality proofs, Bull. Austral. Math. Soc., 22, 431–438.
  11. Beukers, F. (1981) Padé approximations in number theory, Padé approximation and its applications, (Amsterdam, 1980), 90–99, Lecture Notes in Math., Springer, Berlin-New York, 888.
  12. Beukers, F. (1995) Consequences of Apéry’s work on ζ (3), preprint of talk presented at the Rencontres Arithmétiques de Caen, ζ (3), Irrationnel: Les Retombées.
  13. Borwein, J. M. & Bradley, D. M. (1997) Empirically determined Apéry-like formulae for ζ (4n + 3), Experiment. Math., 6 (3), 181–194.
  14. Borwein, J. M., Broadhurst, D. J. & Kamnitzer, J. (2001) Central binomial sums, multiple Clausen values, and zeta Values, Experiment. Math., 10, 25–41.
  15. Cohen, H. (1978) Demonstration de l’irrationalite de ζ (3) (d’aprés R. Apéry), Séminaire de Théorie des Nombres, Grenoble, VI.1–VI.9.
  16. Cohen, J. & Guy, R. (1997) The Book of Numbers, Copernicus, Springer-Verlag New York, Inc.
  17. Gasper, G. & Rahman, M. (2004) Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications), Cambridge Univ. Press, Cambridge.
  18. Gutnik, L. A. (1983) On the irrationality of certain quantities involving ζ (3), Acta Arith., 42, 255–264.
  19. Jones,W. B. & Thorn,W. J. (1980) Continued Fractions Analytic Theory and Applications, In: Encyclopedia of Mathematics and its Applications, Addison-Wesley, London.
  20. Koekoek, R. & Swarttouw, R. F. (1998) The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology.
  21. Nesterenko, Y. V. (1996) A few remarks on ζ (3), Math. Notes, 59 (6), 625–636.
  22. Nesterenko, Y. V. (2003) Integral identities and constructions of approximations to zeta values, J. Théor. Nombres Bordeaux, 15, 535–550.
  23. Nikiforov, A. F. & Uvarov, V. B. (1988) Special Functions in Mathematical Physics, Birkhauser Verlag, Basel.
  24. Perron, O. (1910) Uber ein Satz des Herrn Poincaré, J. Reine Angew. Math., U¨ ber die Poincarésche lineare Differenzgleichung, 137, 6–64.
  25. Petkovsek, M., Wilf, H. S. & Zeilberger, D. (1997) A = B, A. K. Peters, Ltd., Wellesley, M.A.
  26. Pilehrood, H. & Pilehrood, T. H. (2010) Generating function identities for ζ (2n+2), ζ (2n+ 3) via the WZ method, Electron. J. Combin., 15, 223–236.
  27. Pilehrood, H. & Pilehrood, T. H. (2009) Series acceleration formulas for beta values, Discrete Math. Theoret. Comput. Sci., 12 (2), 1–9.
  28. Poincaré, H. (1885) Sur les équations linéaires aux différentielles et aux différences finies, Amer. J. Math., 7, 203–258.
  29. Prévost, M. (1996) A new proof of the irrationality of ζ (2) and ζ (3) using Padé approximants, J. Comput. Appl. Math., 67, 219–235.
  30. Ram, M. (2008) Transcendental numbers and zeta functions, Math. Student., 77 (4), 45–58.
  31. Raymond, A. (1974) Euler and the zeta function, Amer. Math. Monthly, 81, 1067–1087.
  32. Rivoal, T. (2002) Irrationalité d’au moins un des neuf nombres ζ (5), ζ (7), …, ζ (21), Acta Arith., 103 (2), 157–167.
  33. Soria Lorente, A. (2014) Arithmetic of the values of de Riemann’s zeta function in integer arguments, Revista de investigaci´on, G.I.E Pensamiento Matemático, ISSN 2174-0410, IV, 033–0044, 2014.
  34. Soria Lorente, A. (2014) Nesterenko-like rational function, useful to prove the Apéry’s theorem, Notes Number Theory Discrete Math., 20 (2), 79–91.
  35. Sorokin, V. N. (1993) Hermite-Padé approximations for Nikishin systems and the irrationality of (3), Communications of the Moscow Math. Soc., 49, 176–177.
  36. Sorokin, V. N. (1998) Apéry’s theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], 3, 48–52.
  37. Spinade, W. (2003) La familia de número metálicos, Cuadernos del CIMBAGE, 6, 17–44.
  38. Takaaki, M. (2014) Negation of the conjecture for odd zeta values, International Journal of Pure and Applied Mathematics, 91 (1), 103–111.
  39. Van Assche, W. (1999) Multiple orthogonal polynomials, irrationality and transcendence, Contemp. Math., 236, 325–342.
  40. Van Assche, W. (2010) Hermite-Padé Rational Approximation to Irrational Numbers, Computational Methods and Function Theory, 10 (2), 585–602.
  41. Van der Poorten, A. (1978/79) A proof that Euler missed… Apery’s proof of the irrationality of ζ (3), Math. Intelligencer, 1, 195–203.
  42. Zudilin, W. (2001) One of the eight numbers ζ (5), ζ (7), …, ζ (17), ζ (19) is irrational, Mat. Zametki [Math. Notes], 70 (3), 472–476.
  43. Zudilin, W. (2001) Irrationality of values of zeta function at odd integers, Uspekhi Mat. Nauk [Russian Math. Surveys], 56 (2), 215–216.
  44. Zudilin, W. (2001) One of the four numbers ζ (5), ζ (7), ζ (9), ζ (11) is irrational, Uspekhi Mat. Nauk [Russian Math. Surveys], 56 (4), 149–150.
  45. Zudilin,W. (2002) A third-order Apéry-like recursion for ζ (5), Mat. Zametki [Math. Notes], 72 (5), 796–800.
  46. Zudilin, W. (2003) Well-poised hypergeometric service for diophantine problems of zeta values, J. Théor. Nombres Bordeaux, 15 (2), 593–626.

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Cite this paper

Soria Lorente, A. (2018). On Zudilin-like rational approximations to ζ(5). Notes on Number Theory and Discrete Mathematics, 24(2), 104-116, doi: 10.7546/nntdm.2018.24.2.104-116.

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