A modular supercongruence for 6 F 5 : An Apéry-like story  [ Une supercongruence modulaire pour 6 F 5  : un conte à la Apéry ]
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 : 2017-02-05
Accepté le : 2017-11-13
Publié le : 2018-11-22
DOI : https://doi.org/10.5802/aif.3201
Classification : 11B65,  33C20,  33F10
Mots clés : supercongruence, nombres d’Apéry, nombres de type Apéry, fonction hypergéométrique
@article{AIF_2018__68_5_1987_0,
     author = {Osburn, Robert and Straub, Armin and Zudilin, Wadim},
     title = {A modular supercongruence for <span class="mathjax-formula">$\_6F\_5$</span>: An Ap\'ery-like~story},
     journal = {Annales de l'Institut Fourier},
     pages = {1987--2004},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {5},
     year = {2018},
     doi = {10.5802/aif.3201},
     language = {en},
     url = {archive.numdam.org/item/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. http://archive.numdam.org/item/AIF_2018__68_5_1987_0/

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