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Robert Osburn; Armin Straub; Wadim Zudilin
A 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
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Class. Math.: 11B65, 33C20, 33F10
Keywords: supercongruence, Apéry numbers, Apéry-like numbers, hypergeometric function

Résumé - Abstract

We 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.

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