Error Bounds for Asymptotic Expansions of the Ratio of Two Gamma Functions

Error bounds are obtained for asymptotic expansions of the ratio of two gamma functions ${{\Gamma (x + a)} / {\Gamma (x + b)}}$ for the case of real, bounded $a,b$ and large positive x. In particular an assertion of Luke about a result of Fields is rigorously justified by showing that the error made in truncating Fields’ asymptotic expansion is numerically less than and has the same sign as the first neglected term. Use is made of some results for completely monotonic functions and enveloping series.