## A modular supercongruence for $${}_6 F_5$$: an Apéry-like story. (Une supercongruence modulaire pour $${}_6 F_5$$: un conte à la Apéry.)(English. French summary)Zbl 1429.11039

Summary: We prove a supercongruence modulo $$p^3$$ between the $$p$$th 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 $$\zeta(3)$$ 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.

### MSC:

 11B65 Binomial coefficients; factorials; $$q$$-identities 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.) 11F30 Fourier coefficients of automorphic forms

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### References:

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