no. 5
A modular supercongruence for 6 F 5 : An Apéry-like story
[Une supercongruence modulaire pour 6 F 5  : un conte à la Apéry]
Osburn, Robert1 ; Straub, Armin2 ; Zudilin, Wadim3, 4
1 School of Mathematics and Statistics University College Dublin Belfield, Dublin 4 (Ireland)
2 Dept. of Mathematics and Statistics University of South Alabama 411 University Blvd N MSPB 325, Mobile, AL 36688 (USA)
3 Dept. of Mathematics, IMAPP Radboud Universiteit PO Box 9010 6500 GL Nijmegen (Netherlands)
4 School of Math. and Phys. Sciences The University of Newcastle Callaghan, NSW 2308 (Australia)
Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 1987-2004.

On démontre une supercongruence modulo p 3 entre le p-ième coefficient de Fourier d’une forme modulaire de poids 6 et une série hypergéométrique 6 F 5 tronquée. Les nouveaux ingrédients de la preuve sont la comparaison de deux approximations rationnelles de ζ(3) pour produire des identités non triviales entre sommes harmoniques, et la réduction des congruences qui en résultent entre des sommes via une congruence qui relie les nombres d’Apéry á une autre suite du type de celle d’Apéry.

We prove a supercongruence modulo p 3 between the pth Fourier coefficient of a weight 6 modular form and a truncated 6 F 5 -hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to ζ(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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3201
Classification : 11B65, 33C20, 33F10
Keywords: supercongruence, Apéry numbers, Apéry-like numbers, hypergeometric function
Mot clés : supercongruence, nombres d’Apéry, nombres de type Apéry, fonction hypergéométrique

Osburn, Robert 1 ; Straub, Armin 2 ; Zudilin, Wadim 3, 4

1 School of Mathematics and Statistics University College Dublin Belfield, Dublin 4 (Ireland)
2 Dept. of Mathematics and Statistics University of South Alabama 411 University Blvd N MSPB 325, Mobile, AL 36688 (USA)
3 Dept. of Mathematics, IMAPP Radboud Universiteit PO Box 9010 6500 GL Nijmegen (Netherlands)
4 School of Math. and Phys. Sciences The University of Newcastle Callaghan, NSW 2308 (Australia)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2018__68_5_1987_0,
     author = {Osburn, Robert and Straub, Armin and Zudilin, Wadim},
     title = {A modular supercongruence for $_6F_5$: {An} {Ap\'ery-like~story}},
     journal = {Annales de l'Institut Fourier},
     pages = {1987--2004},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {5},
     year = {2018},
     doi = {10.5802/aif.3201},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3201/}
}
TY  - JOUR
AU  - Osburn, Robert
AU  - Straub, Armin
AU  - Zudilin, Wadim
TI  - A modular supercongruence for $_6F_5$: An Apéry-like story
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 1987
EP  - 2004
VL  - 68
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3201/
DO  - 10.5802/aif.3201
LA  - en
ID  - AIF_2018__68_5_1987_0
ER  - 
%0 Journal Article
%A Osburn, Robert
%A Straub, Armin
%A Zudilin, Wadim
%T A modular supercongruence for $_6F_5$: An Apéry-like story
%J Annales de l'Institut Fourier
%D 2018
%P 1987-2004
%V 68
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3201/
%R 10.5802/aif.3201
%G en
%F AIF_2018__68_5_1987_0
Osburn, Robert; Straub, Armin; Zudilin, Wadim. A modular supercongruence for $_6F_5$: An Apéry-like story. Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 1987-2004. doi : 10.5802/aif.3201. https://aif.centre-mersenne.org/articles/10.5802/aif.3201/

[1] Ahlgren, Scott; Ono, Ken A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math., Volume 518 (2000), pp. 187-212 | DOI | MR | Zbl

[2] Apéry, Roger Irrationalité de ζ(2) et ζ(3) (Astérisque), Volume 61, Société Mathématique de France, 1979, pp. 11-13 | Zbl

[3] Bailey, Wilfrid Norman Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, 32, Stechert-Hafner, 1964, v+108 pages | MR

[4] Beukers, Frits Another congruence for the Apéry numbers, J. Number Theory, Volume 25 (1987) no. 2, pp. 201-210 | DOI | MR | Zbl

[5] Beukers, Frits Irrationality proofs using modular forms, Journées arithmétiques de Besançon (Besançon, 1985) (Astérisque), Volume 147-148, Société Mathématique de France, 1987, pp. 271-283 | MR | Zbl

[6] Chu, Wenchang; De Donno, Livia Hypergeometric series and harmonic number identities, Adv. Appl. Math., Volume 34 (2005) no. 1, pp. 123-137 | DOI | MR | Zbl

[7] Cooper, Shaun Sporadic sequences, modular forms and new series for 1/π, Ramanujan J., Volume 29 (2012) no. 1-3, pp. 163-183 | DOI | MR | Zbl

[8] Frechette, Sharon; Ono, Ken; Papanikolas, Matthew Gaussian hypergeometric functions and traces of Hecke operators, Int. Math. Res. Not., Volume 2004 (2004) no. 60, pp. 3233-3262 | DOI | MR | Zbl

[9] Fuselier, Jenny G.; Long, Ling; Ramakrishna, Ravi; Swisher, Holly; Tu, Fang-Ting Hypergeometric functions over finite fields (2015) (http://arxiv.org/abs/1510.02575)

[10] Fuselier, Jenny G.; McCarthy, Dermot Hypergeometric type identities in the p-adic setting and modular forms, Proc. Am. Math. Soc., Volume 144 (2016) no. 4, pp. 1493-1508 | DOI | MR | Zbl

[11] Greene, John Hypergeometric functions over finite fields, Trans. Am. Math. Soc., Volume 301 (1987) no. 1, pp. 77-101 | DOI | MR | Zbl

[12] van Hamme, L. Some conjectures concerning partial sums of generalized hypergeometric series, p-adic functional analysis (Nijmegen, 1996) (Lecture Notes in Pure and Appl. Math.), Volume 192, Dekker, 1997, pp. 223-236 | MR | Zbl

[13] Kibelbek, Jonas; Long, Ling; Moss, Kevin; Sheller, Benjamin; Yuan, Hao Supercongruences and complex multiplication, J. Number Theory, Volume 164 (2016), pp. 166-178 | DOI | MR | Zbl

[14] Kilbourn, Timothy An extension of the Apéry number supercongruence, Acta Arith., Volume 123 (2006) no. 4, pp. 335-348 | DOI | MR | Zbl

[15] Krattenthaler, Christian; Rivoal, Tanguy Hypergéométrie et fonction zêta de Riemann, Mem. Am. Math. Soc., Volume 186 (2007) no. 875, x+87 pages | DOI | MR | Zbl

[16] McCarthy, Dermot Binomial coefficient-harmonic sum identities associated to supercongruences, Integers, Volume 11 (2011), A37 (Art A37, 8 p.) | DOI | MR | Zbl

[17] McCarthy, Dermot Extending Gaussian hypergeometric series to the p-adic setting, Int. J. Number Theory, Volume 8 (2012) no. 7, pp. 1581-1612 | DOI | MR | Zbl

[18] McCarthy, Dermot On a supercongruence conjecture of Rodriguez-Villegas, Proc. Am. Math. Soc., Volume 140 (2012) no. 7, pp. 2241-2254 | DOI | MR | Zbl

[19] Nesterenko Some remarks on ζ(3), Mat. Zametki, Volume 59 (1996) no. 6, pp. 865-880 | DOI | MR | Zbl

[20] Osburn, Robert; Schneider, Carsten Gaussian hypergeometric series and supercongruences, Math. Comput., Volume 78 (2009) no. 265, pp. 275-292 | DOI | MR | Zbl

[21] Osburn, Robert; Zudilin, Wadim On the (K.2) supercongruence of Van Hamme, J. Math. Anal. Appl., Volume 433 (2016) no. 1, pp. 706-711 | DOI | MR | Zbl

[22] Paule, Peter; Schneider, Carsten Computer proofs of a new family of harmonic number identities, Adv. Appl. Math., Volume 31 (2003) no. 2, pp. 359-378 | DOI | MR | Zbl

[23] Petkovšek, Marko; Wilf, Herbert S.; Zeilberger, Doron A=B, Peters, 1996, xii+212 pages (With a foreword by Donald E. Knuth, With a separately available computer disk) | MR | Zbl

[24] van der Poorten, Alfred A proof that Euler missed: Apéry’s proof of the irrationality of ζ(3), Math. Intell., Volume 1 (1979) no. 4, pp. 195-203 | DOI | MR | Zbl

[25] Rivoal, Tanguy Propriétés diophantinnes des valeurs de la fonction zêta de Riemann aux entiers impairs, Université de Caen (France) (2001) (Ph. D. Thesis)

[26] Rodriguez-Villegas, Fernando Hypergeometric families of Calabi–Yau manifolds, Calabi–Yau varieties and mirror symmetry (Toronto, ON, 2001) (Fields Inst. Commun.), Volume 38, American Mathematical Society, 2003, pp. 223-231 | MR | Zbl

[27] Schneider, Carsten Symbolic summation assists combinatorics, Sémin. Lothar. Comb., Volume 56 (2007), B56b http://www.mat.univie.ac.at/slc/wpapers/s56schneider.html (Art. B56b, 36 p.) | Zbl

[28] Sloane, Neil J. A. The On-Line Encyclopedia of Integer Sequences, 2017 (published electronically at http://oeis.org)

[29] Swisher, Holly On the supercongruence conjectures of van Hamme, Res. Math. Sci., Volume 2 (2015), 18 (Art. 18, 21 p.) | DOI | MR | Zbl

[30] Zagier, Don Integral solutions of Apéry-like recurrence equations, Groups and symmetries (CRM Proc. Lecture Notes), Volume 47, American Mathematical Society, 2009, pp. 349-366 | MR | Zbl

[31] Zudilin, Wadim Apéry’s theorem. Thirty years after, Int. J. Math. Comput. Sci., Volume 4 (2009) no. 1, pp. 9-19 | MR | Zbl

[32] Zudilin, Wadim A generating function of the squares of Legendre polynomials, Bull. Aust. Math. Soc., Volume 89 (2014) no. 1, pp. 125-131 | DOI | MR | Zbl

[33] Zudilin, Wadim Hypergeometric heritage of W. N. Bailey. With an appendix: Bailey’s letters to F. Dyson (2016) (http://arxiv.org/abs/1611.08806)

Cité par Sources :