The analytic continuation of the Gaussian hypergeometric function 2F1( a, b; c; z) for arbitrary parameters

Abstract

Our objective is to provide a complete table of analytic condinuation formulas for the Gaussian hypergeometric function 2 F 1( a, b; c; z) which allow its fast and accurate computation for arbitrary values of z and of the parameters a, b, c. To this end we distinguish 12 basis sets of the two-dimensional space of the solutions of the hypergeometric differential equation. Representing 2 F 1 in each of them yields 12 analytic continuation formulas. Each two of them are series in one of the arguments z, z/(z−1), (1−z), (1−1/z), 1/z, 1/(1−z) , respectively, such that any given argument z, with the exception of two single points in the complex plane, lies in the convergence domain of at least one of them. We present rapidly converging series representations of 2 F 1 for all possible constellations of parameters. For minimizing the effort for the derivation of these series we have extensively used the symmetry group of the hypergeometric equation, which is shown to be isomorphic to the cubic group O h .