Asymptotic expansions for the gamma function in terms of hyperbolic functions

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.