A summation formula and Ramanujan type series

DOI10.1016/j.jmaa.2011.12.048zbMath1273.33007OpenAlexW2063506711MaRDI QIDQ764949

Zhi-Guo Liu

Publication date: 16 March 2012

Published in: Journal of Mathematical Analysis and Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.jmaa.2011.12.048

zbMATH Keywords

gamma functionhypergeometric functionsDougall's \(_5F_4\) summationRamanujan type series

Mathematics Subject Classification ID

Gamma, beta and polygamma functions (33B15) Generalized hypergeometric series, ({}_pF_q) (33C20) Evaluation of number-theoretic constants (11Y60)

Related Items (11)

Double series expansions for \(\pi\) ⋮ From Wallis and Forsyth to Ramanujan ⋮ GAUSS SUMMATION AND RAMANUJAN-TYPE SERIES FOR 1/π ⋮ Extensions of Ramanujan's two formulas for \(1/\pi\) ⋮ Unnamed Item ⋮ Hypergeometric series summations and \(\pi\)-formulas ⋮ \(\pi\)-formulas with free parameters ⋮ $q$-analogues of several $\pi $-formulas ⋮ Series expansions for \(1/\pi^m\) and \(\pi^m\) ⋮ Extensions of the classical theorems for very well-poised hypergeometric functions ⋮ Two hypergeometric summation theorems and Ramanujan-type series

Cites Work

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• A new Ramanujan-like series for \(1/\pi ^{2}\)
• \(\pi \)-formulas implied by Dougall's summation theorem for \(_{5} F _{4}\)-series
• Eisenstein series and Ramanujan-type series for \(1 / \pi\)
• The Apéry numbers, the Almkvist-Zudilin numbers and new series for \(1/\pi\)
• New \(_{5}F_{4}\) hypergeometric transformations, three-variable Mahler measures, and formulas for \(1/ \pi \)
• Ramanujan's Eisenstein series and new hypergeometric-like series for \(1/\pi ^{2}\)
• Class number three Ramanujan type series for \(1/\pi\)
• Domb's numbers and Ramanujan-Sato type series for \(1/\pi\)
• Hypergeometric identities for 10 extended Ramanujan-type series
• Ramanujan's series for \(1/\pi \) arising from his cubic and quartic theories of elliptic functions
• New hypergeometric-like series for $1/\pi^{2}$ arising from Ramanujan’s theory of elliptic functions to alternative base 3
• Dougall’s bilateral ₂𝐻₂-series and Ramanujan-like 𝜋-formulae
• Ramanujan's Series for 1/π: A Survey
• RAMANUJAN'S CLASS INVARIANT λ_n AND A NEW CLASS OF SERIES FOR 1/π
• GAUSS SUMMATION AND RAMANUJAN-TYPE SERIES FOR 1/π
• NEW REPRESENTATIONS FOR APÉRY‐LIKE SEQUENCES
• Ramanujan's cubic continued fraction revisited
• More Ramanujan-type formulae for $ 1/\pi^2$
• Series and iterations for 1/π
• The Rogers-Ramanujan continued fraction and a quintic iteration for $1/\pi$
• Using Fourier-Legendre expansions to derive series for \(\frac{1}{\pi}\) and \(\frac{1}{\pi^{2}}\)

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