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Accelerating the Hypergeometric Function with the Beta Integral to Derive New Infinite Series for $\pi$ and Values of the Gamma Function
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abstract
The beta integral is applied to accelerate the hypergeometric function $2 F 1\left\{1, B; C ; w\right\}$ to derive new infinite series for constants such as $\pi$ and values of the gamma function. A compendium of new infinite series is given. Ramanujan-like formulas for pi are also derived based on elementary inverse trigonometric functions, including a formula with rational values that adds 2.5 digits per terms, which makes the series much more compact than similar formulas in the existing literature.
Forward citations
Cited by 3 Pith papers
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Analytic Dilogarithm Identities
A beta integral method yields analytic proofs of conjectured dilogarithm identities and new ladder relations with quartic and sextic bases.
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Fast Ramanujan-type Series for Logarithms. Part I
The paper introduces new, mostly proven hypergeometric series for log 2, log 3, and log 5 with lower binary splitting costs than Machin-type formulas, plus a variable-p family.
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Functional Dilogarithm Identities in Quadratic Fields
Ratio of sextic and cubic arctangent integrals equals a rational constant and yields new dilogarithm functional equations plus proofs of several prior conjectures.
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