We prove a supercongruence modulo between the th Fourier coefficient of a weight 6 modular form and a truncated -hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to 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 entre le -ième coefficient de Fourier d’une forme modulaire de poids 6 et une série hypergéométrique tronquée. Les nouveaux ingrédients de la preuve sont la comparaison de deux approximations rationnelles de 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.
Accepted:
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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
@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, 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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