Uniform asymptotic expansions for hypergeometric functions with large parameters IV

Abstract

In this paper, we obtain asymptotic expansions for the Gauss hypergeometric function [Formula: see text], where ej = 0, ± 1, j = 1, 2, 3, as |λ| → ∞. We complete the results of three previous publications [Uniform asymptotic expansions for hypergeometric functions with large parameters I, Anal. Appl. (Singap.) 1 (2003) 111–120; Uniform asymptotic expansions for hypergeometric functions with large parameters II, Anal. Appl. (Singap.) 1 (2003) 121–128; Uniform asymptotic expansions for hypergeometric functions with large parameters III, Anal. Appl. (Singap.) 8 (2010) 199–210], discuss all cases and, what is new, we consider now all critical values of z. For one case, the full details of the well-known Bleistein method are given, since a new technical detail is observed.