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Osburn, Robert; Straub, Armin; Zudilin, Wadim

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

Ann. Inst. Fourier 68, No. 5, 1987-2004 (2018).
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.

Cited in 10 Documents

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

Keywords:

supercongruence; Apéry numbers; Apéry-like numbers; hypergeometric function

Software:

OEIS; SIGMA
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\textit{R. Osburn} et al., Ann. Inst. Fourier 68, No. 5, 1987--2004 (2018; Zbl 1429.11039)
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Online Encyclopedia of Integer Sequences:

a(n) = Sum_{k = 0..n} binomial(n,k)^4.
Central terms of triangle A181544.

References:

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