no. 5
A modular supercongruence for 6 F 5 : An Apéry-like story
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, Volume 68 (2018) no. 5, pp. 1987-2004.

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.

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.

Received:
Accepted:
Published online:
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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Osburn, Robert; Straub, Armin; Zudilin, Wadim. A modular supercongruence for $_6F_5$: An Apéry-like story. Annales de l'Institut Fourier, Volume 68 (2018) no. 5, pp. 1987-2004. doi : 10.5802/aif.3201. https://aif.centre-mersenne.org/articles/10.5802/aif.3201/

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