With cedram.org English français Home Overview Search for an article Submit a paper Informations for the authors Subscription Limited access - RUCHE Table of contents for this issue | Previous article | Next article Robert Osburn; Armin Straub; Wadim ZudilinA modular supercongruence for $_6F_5$: An Apéry-like story(Une supercongruence modulaire pour $_6F_5$ : un conte à la Apéry)Annales de l'institut Fourier, 68 no. 5 (2018), p. 1987-2004, doi: 10.5802/aif.3201 Article PDF Class. Math.: 11B65, 33C20, 33F10Keywords: supercongruence, Apéry numbers, Apéry-like numbers, hypergeometric function Résumé - AbstractWe prove a supercongruence modulo $p^3$ between the $p$th Fourier coefficient of a weight 6 modular form and a truncated ${}_6F_5$-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to $\zeta (3)$ to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence relating the Apéry numbers to another Apéry-like sequence. Bibliography[1] Scott Ahlgren & Ken Ono, “A Gaussian hypergeometric series evaluation and Apéry number congruences”, J. Reine Angew. Math. 518 (2000), p. 187-212 Article[2] Roger Apéry, Irrationalité de $\zeta (2)$ et $\zeta (3)$, Astérisque 61, Société Mathématique de France, 1979, p. 11–13 [3] Wilfrid Norman Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics 32, Stechert-Hafner, 1964 [4] Frits Beukers, “Another congruence for the Apéry numbers”, J. Number Theory 25 (1987) no. 2, p. 201-210 Article[5] Frits Beukers, Irrationality proofs using modular forms, Journées arithmétiques de Besançon (Besançon, 1985), Astérisque 147-148, Société Mathématique de France, 1987, p. 271–283 [6] Wenchang Chu & Livia De Donno, “Hypergeometric series and harmonic number identities”, Adv. Appl. Math. 34 (2005) no. 1, p. 123-137 Article[7] Shaun Cooper, “Sporadic sequences, modular forms and new series for $1/\pi$”, Ramanujan J. 29 (2012) no. 1-3, p. 163-183 Article[8] Sharon Frechette, Ken Ono & Matthew Papanikolas, “Gaussian hypergeometric functions and traces of Hecke operators”, Int. Math. Res. Not. 2004 (2004) no. 60, p. 3233-3262 Article[9] Jenny G. Fuselier, Ling Long, Ravi Ramakrishna, Holly Swisher & Fang-Ting Tu, “Hypergeometric functions over finite fields”, http://arxiv.org/abs/1510.02575, 2015 [10] Jenny G. Fuselier & Dermot McCarthy, “Hypergeometric type identities in the $p$-adic setting and modular forms”, Proc. Am. Math. Soc. 144 (2016) no. 4, p. 1493-1508 Article[11] John Greene, “Hypergeometric functions over finite fields”, Trans. Am. Math. Soc. 301 (1987) no. 1, p. 77-101 Article[12] L. van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, $p$-adic functional analysis (Nijmegen, 1996), Lecture Notes in Pure and Appl. Math. 192, Dekker, 1997, p. 223–236 [13] Jonas Kibelbek, Ling Long, Kevin Moss, Benjamin Sheller & Hao Yuan, “Supercongruences and complex multiplication”, J. Number Theory 164 (2016), p. 166-178 Article[14] Timothy Kilbourn, “An extension of the Apéry number supercongruence”, Acta Arith. 123 (2006) no. 4, p. 335-348 Article[15] Christian Krattenthaler & Tanguy Rivoal, “Hypergéométrie et fonction zêta de Riemann”, Mem. Am. Math. Soc. 186 (2007) no. 875 Article[16] Dermot McCarthy, “Binomial coefficient-harmonic sum identities associated to supercongruences”, Integers 11 (2011), Art A37, 8 p. Article[17] Dermot McCarthy, “Extending Gaussian hypergeometric series to the $p$-adic setting”, Int. J. Number Theory 8 (2012) no. 7, p. 1581-1612 Article[18] Dermot McCarthy, “On a supercongruence conjecture of Rodriguez-Villegas”, Proc. Am. Math. Soc. 140 (2012) no. 7, p. 2241-2254 Article[19] Nesterenko, “Some remarks on $\zeta (3)$”, Mat. Zametki 59 (1996) no. 6, p. 865-880 Article[20] Robert Osburn & Carsten Schneider, “Gaussian hypergeometric series and supercongruences”, Math. Comput. 78 (2009) no. 265, p. 275-292 Article[21] Robert Osburn & Wadim Zudilin, “On the (K.2) supercongruence of Van Hamme”, J. Math. Anal. Appl. 433 (2016) no. 1, p. 706-711 Article[22] Peter Paule & Carsten Schneider, “Computer proofs of a new family of harmonic number identities”, Adv. Appl. Math. 31 (2003) no. 2, p. 359-378 Article[23] Marko Petkovšek, Herbert S. Wilf & Doron Zeilberger, $A=B$, Peters, 1996, With a foreword by Donald E. Knuth, With a separately available computer disk [24] Alfred van der Poorten, “A proof that Euler missed: Apéry’s proof of the irrationality of $\zeta (3)$”, Math. Intell. 1 (1979) no. 4, p. 195-203 Article[25] Tanguy Rivoal, Propriétés diophantinnes des valeurs de la fonction zêta de Riemann aux entiers impairs, Ph. D. Thesis, Université de Caen (France), 2001 [26] Fernando Rodriguez-Villegas, Hypergeometric families of Calabi–Yau manifolds, Calabi–Yau varieties and mirror symmetry (Toronto, ON, 2001), Fields Inst. Commun. 38, American Mathematical Society, 2003, p. 223–231 [27] Carsten Schneider, “Symbolic summation assists combinatorics”, Sémin. Lothar. Comb. 56 (2007), Art. B56b, 36 p. [28] Neil J. A. Sloane, “The On-Line Encyclopedia of Integer Sequences” 2017, published electronically at http://oeis.org [29] Holly Swisher, “On the supercongruence conjectures of van Hamme”, Res. Math. Sci. 2 (2015), Art. 18, 21 p. Article[30] Don Zagier, Integral solutions of Apéry-like recurrence equations, Groups and symmetries, CRM Proc. Lecture Notes 47, American Mathematical Society, 2009, p. 349–366 [31] Wadim Zudilin, “Apéry’s theorem. Thirty years after”, Int. J. Math. Comput. Sci. 4 (2009) no. 1, p. 9-19 [32] Wadim Zudilin, “A generating function of the squares of Legendre polynomials”, Bull. Aust. Math. Soc. 89 (2014) no. 1, p. 125-131 Article[33] Wadim Zudilin, “Hypergeometric heritage of W. N. Bailey. With an appendix: Bailey’s letters to F. Dyson”, http://arxiv.org/abs/1611.08806, 2016 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310