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On hypergeometric identities related to zeta values. (English) Zbl 1429.11122

Summary: Two linear forms, \(\sigma _n \zeta (5)+\tau _n \zeta (3)+\varphi _n\) and \(\sigma _n\zeta (2)+\tau _n/2\), with suitable rational coefficients \(\sigma _n,\tau _n,\varphi _n\), are presented. As a byproduct, we obtain an identity between simple and double binomial sums, where the simple sum is the value of a terminating well-poised Saalschützian \(_4 F_3\) series. This complements a recent note of the author [Mosc. J. Comb. Number Theory 3, No. 3-4, 145–168 (2013; Zbl 1352.11059)] on two linear forms: \(\alpha _n \widetilde{\zeta }(4)+\beta _n \widetilde{\zeta }(2)+\gamma _n\), based on an identity of Paule-Schneider, and \(\alpha _n\zeta (2)+\beta _n\), coming from the Apéry-Beukers construction.

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
05A19 Combinatorial identities, bijective combinatorics
11B65 Binomial coefficients; factorials; \(q\)-identities
33C20 Generalized hypergeometric series, \({}_pF_q\)

Citations:

Zbl 1352.11059
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References:

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