Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal

Abstract

In (Boyd, Proc. R. Soc. Lond. A 447 (1994) 609--630), W. G. C. Boyd derived a resurgence representation for the gamma function, exploiting the reformulation of the method of steepest descents by M. Berry and C. Howls (Berry and Howls, Proc. R. Soc. Lond. A 434 (1991) 657--675). Using this representation, he was able to derive a number of properties of the asymptotic expansion for the gamma function, including explicit and realistic error bounds, the smooth transition of the Stokes discontinuities, and asymptotics for the late coefficients. The main aim of this paper is to modify the resurgence formula of Boyd making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal. We also prove the exponentially improved versions of these expansions complete with error terms. Finally, we provide new (formal) asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors.