Asymptotic expansions of gamma and related functions, binomial coefficients, inequalities and means

Abstract

We give an overview of the use of asymptotic expansions of gamma and related functions --- ratio of gamma functions, powers, digamma and polygamma functions. The aim is to introduce a general theory which can unify various particular formulas for factorial functions and binomial coefficients. The connection with inequalities for gamma function is established. Also, a systematic approach to asymptotic expansion of various integral means, bivariate classical and parameter means is given, with applications to comparison of means.