Uniform convergent expansions of the Gauss hypergeometric function in terms of elementary functions

This is an accepted manuscript of an article published by Taylor & Francis in Integral Transforms and Special Functions on 2018-09-28, available online: https://doi.org/10.1080/10652469.2018.1525369We consider the hypergeometric function 2F1(a, b; c; z) for z ∈ C \ [1,∞). For Ra ≥ 0, we derive a convergent expansion of 2F1(a, b; c; z) in terms of the function (1 − z)−a and of rational functions of z that is uniformly valid for z in any compact in C \ [1,∞). When a ∈ N, the expansion also contains a logarithmic term of the form log(1 − z). For Ra ≤ 0, we derive a convergent expansion of (1 − z)a 2F1(a, b; c; z) in terms of the function (1 − z)−a and of rational functions of z that is uniformly valid for z in any compact in C \ [1,∞) in the exterior of the circle |z − 1| = r for arbitrary r > 0. The expansions are accompanied by realistic error bounds. Some numerical experiments show the accuracy of the approximation.This research was supported by Ministerio de Economía, Industria y Competitividad, Gobierno de España (MTM2017-83490-P)