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
Download full paper: PDF, 209 Kb
Details
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
- 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.
- Abramov, S. A. & Le, H. Q. (2002) A criterion for the applicability of Zeilberger’s algorithm to rational functions, Discrete Math., 259, 1–17.
- Abramov, S. A. (2003) When does Zeilberger’s algorithm succeed?, Appl. Math., 30, 424– 441, 2003.
- 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.
- Abramowitz, M. & Stegun, I. A. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York.
- Almkvist, G. & Granville, A. (1999) Borwein and Bradley’s Apéry-like formulae for ζ (4n + 3), Experiment. Math., 8, 197–203.
- Apéry, R. (1979) Irrationalité de ζ (2) et ζ (3), Astérisque, 61, 11–13.
- Arvesú, J. (2012) Orthogonal forms: A key tool for deducing Apéry’s recurrence relation, J. Approx. Theory, accepted, 2012.
- Beukers, F. (1979) A note on the irrationality of ζ (2) and ζ (3), Bull. London Math. Soc., 11, 268–272.
- Beukers, F. (1980) Legendre polynomials in irrationality proofs, Bull. Austral. Math. Soc., 22, 431–438.
- 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.
- 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.
- Borwein, J. M. & Bradley, D. M. (1997) Empirically determined Apéry-like formulae for ζ (4n + 3), Experiment. Math., 6 (3), 181–194.
- Borwein, J. M., Broadhurst, D. J. & Kamnitzer, J. (2001) Central binomial sums, multiple Clausen values, and zeta Values, Experiment. Math., 10, 25–41.
- Cohen, H. (1978) Demonstration de l’irrationalite de ζ (3) (d’aprés R. Apéry), Séminaire de Théorie des Nombres, Grenoble, VI.1–VI.9.
- Cohen, J. & Guy, R. (1997) The Book of Numbers, Copernicus, Springer-Verlag New York, Inc.
- Gasper, G. & Rahman, M. (2004) Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications), Cambridge Univ. Press, Cambridge.
- Gutnik, L. A. (1983) On the irrationality of certain quantities involving ζ (3), Acta Arith., 42, 255–264.
- Jones,W. B. & Thorn,W. J. (1980) Continued Fractions Analytic Theory and Applications, In: Encyclopedia of Mathematics and its Applications, Addison-Wesley, London.
- 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.
- Nesterenko, Y. V. (1996) A few remarks on ζ (3), Math. Notes, 59 (6), 625–636.
- Nesterenko, Y. V. (2003) Integral identities and constructions of approximations to zeta values, J. Théor. Nombres Bordeaux, 15, 535–550.
- Nikiforov, A. F. & Uvarov, V. B. (1988) Special Functions in Mathematical Physics, Birkhauser Verlag, Basel.
- Perron, O. (1910) Uber ein Satz des Herrn Poincaré, J. Reine Angew. Math., U¨ ber die Poincarésche lineare Differenzgleichung, 137, 6–64.
- Petkovsek, M., Wilf, H. S. & Zeilberger, D. (1997) A = B, A. K. Peters, Ltd., Wellesley, M.A.
- 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.
- Pilehrood, H. & Pilehrood, T. H. (2009) Series acceleration formulas for beta values, Discrete Math. Theoret. Comput. Sci., 12 (2), 1–9.
- Poincaré, H. (1885) Sur les équations linéaires aux différentielles et aux différences finies, Amer. J. Math., 7, 203–258.
- 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.
- Ram, M. (2008) Transcendental numbers and zeta functions, Math. Student., 77 (4), 45–58.
- Raymond, A. (1974) Euler and the zeta function, Amer. Math. Monthly, 81, 1067–1087.
- Rivoal, T. (2002) Irrationalité d’au moins un des neuf nombres ζ (5), ζ (7), …, ζ (21), Acta Arith., 103 (2), 157–167.
- 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.
- Soria Lorente, A. (2014) Nesterenko-like rational function, useful to prove the Apéry’s theorem, Notes Number Theory Discrete Math., 20 (2), 79–91.
- Sorokin, V. N. (1993) Hermite-Padé approximations for Nikishin systems and the irrationality of (3), Communications of the Moscow Math. Soc., 49, 176–177.
- Sorokin, V. N. (1998) Apéry’s theorem, Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], 3, 48–52.
- Spinade, W. (2003) La familia de número metálicos, Cuadernos del CIMBAGE, 6, 17–44.
- Takaaki, M. (2014) Negation of the conjecture for odd zeta values, International Journal of Pure and Applied Mathematics, 91 (1), 103–111.
- Van Assche, W. (1999) Multiple orthogonal polynomials, irrationality and transcendence, Contemp. Math., 236, 325–342.
- Van Assche, W. (2010) Hermite-Padé Rational Approximation to Irrational Numbers, Computational Methods and Function Theory, 10 (2), 585–602.
- 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.
- 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.
- Zudilin, W. (2001) Irrationality of values of zeta function at odd integers, Uspekhi Mat. Nauk [Russian Math. Surveys], 56 (2), 215–216.
- 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.
- Zudilin,W. (2002) A third-order Apéry-like recursion for ζ (5), Mat. Zametki [Math. Notes], 72 (5), 796–800.
- Zudilin, W. (2003) Well-poised hypergeometric service for diophantine problems of zeta values, J. Théor. Nombres Bordeaux, 15 (2), 593–626.
Related papers
- Soria-Lorente, A. & Berres, S. (2020). A single parameter Hermite–Padé series representation for Apéry’s constant. Notes on Number Theory and Discrete Mathematics, 26 (3), 107-134, doi: 10.7546/nntdm.2020.26.3.107-134.
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.