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• W2946688770 abstract "The Smith's approximation formula for the gamma function is given byΓ(x+12)=2π(xe)x(2xtanh⁡12x)x/2(1+O(1x5)), as x→∞. In this paper, by means of a little known power series, we develop this formula to an asymptotic expansions:Γ(x+1/2)2π(x/e)x∼(2xtanh⁡12x)x/2exp⁡(∑n=3∞[(2n)!−(2n−1)22n−1](1−21−2n)2n(2n−1)(2n)!B2nx2n−1), as x→∞, and give an estimate of remainder in the above asymptotic series, where B2n is the Bernoulli number. Moreover, as relevant results, Ramanujan type asymptotic expansions for the gamma function Γ(x+1/2) are obtained; an asymptotic expansion for Wallis ratio and an estimate of the remainder are established." @default.
• W2946688770 created "2019-05-29" @default.
• W2946688770 creator A5021351934 @default.
• W2946688770 creator A5088913538 @default.
• W2946688770 date "2019-10-01" @default.
• W2946688770 modified "2023-10-09" @default.
• W2946688770 title "Asymptotic expansions for the gamma function in terms of hyperbolic functions" @default.
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• W2946688770 doi "https://doi.org/10.1016/j.jmaa.2019.05.022" @default.
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