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Pré-Publication, Document De Travail Année : 2022

Series acceleration formulas obtained from experimentally discovered hypergeometric recursions

Paul Levrie
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Résumé

In 2010, Kh.\ Hessami Pilehrood and T.\ Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $\beta$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences that we have discovered experimentally, and we prove these recurrences using the WZ method, and apply these recurrences to obtain series acceleration identities, including a family of summations generalizing a series for $\frac{1}{\pi^2}$ due to Guillera, and a family of summations generalizing an accelerated series for Catalan's constant due to Lupa\c{s}. We also provide ``non-computer'' proofs of our hypergeometric transforms.
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Dates et versions

hal-03659300 , version 1 (04-05-2022)
hal-03659300 , version 2 (05-11-2022)
hal-03659300 , version 3 (20-12-2022)

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  • HAL Id : hal-03659300 , version 1

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Paul Levrie, John Campbell. Series acceleration formulas obtained from experimentally discovered hypergeometric recursions. 2022. ⟨hal-03659300v1⟩
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