Error bounds and exponential improvement for Hermite's asymptotic expansion for the Gamma function

In this paper we reconsider the asymptotic expansion of the Gamma function with shifted argument, which is the generalization of the well-known Stirling series. To our knowledge, no explicit error bounds exist in the literature for this expansion. Therefore, the first aim of this paper is to extend the known error estimates of Stirling's series to this general case. The second aim is to give exponentially-improved asymptotics for this asymptotic series.